referring now to the drawings and with particular reference to fig1 there is here shown one form of the engine auxiliary balancing device which is part of the present invention . crank 11 is a special crank which is part of the crankshaft 12 ( certain conventional components not shown ) for an internal - combustion engine of the reciprocating type . crank 11 is referred to as a balance crank . the cylindrical journal portion of the balance crank 11 may be referred to as the balance crankpin . the radius of the crank circle for the balance crank 11 is r b . the fixed bearing 13 represents one of the crankshaft main bearings commonly used to support a crankshaft 12 . depending upon engine design , a main bearing may or may not be required at the particular point shown . the rod 14 is a special rod having some similarity to a conventional connecting rod . to distinguish it , however , from a conventional connecting rod , rod 14 is referred to as a balance rod . the member 15 is referred to as a balance rocker . at the approximate junction of the balance rod 14 and the balance rocker 15 is an enlarged mass which is referred to as the balance weight 16 . the balance weight 16 may be a part of the balance rod 14 or balance rocker 15 , or its mass may be divided between the two as indicated in fig1 . at one end of the balance rocker 15 is the balance rocker fixed bearing 17 . the axis of the bearing is also shown , as part of the schematic representation . at the other end of the balance rocker 15 and connecting it to the balance rod 14 is the balance weight bearing 18 . the bearing is designated at its axis , since the actual bearing is obscured in this view by the balance weight 16 . the length of the balance rocker 15 between the axes of the bearings at its ends is designated by r . the length of the balance rod 14 between the axes of the bearings at its ends is designated by l b . the symbol w b will also be used in further discussion . when the center of gravity of the balance weight 16 is on the axis of the balance weight bearing 18 , w b will represent the actual weight of the balance weight 16 . when the center of gravity is not on the axis of the bearing 18 , w b will represent an adjusted weight which can be considered as acting at the axis or at an equivalent radius r from the axis of the guide link fixed bearing 17 . the length r of the balance rocker 15 is , for example , approximately two and one - third to three and one - third times the radius r b of the crank circle for the balance crank 11 . thus , when the crankshaft 12 rotates , as for delivering power from an engine , the balance weight 16 oscillates back and forth in an arc . the three moving &# 34 ; links &# 34 ; having lenghts r b , l b and r together with the engine static structure as a stationary &# 34 ; link &# 34 ;, may be viewed as a form of four - link mechanism or four - bar linkage ( i . e . a crank - and - rocker mechanism ). this general type of linkage is known in the prior art on machinery , and its motion , which is highly complicated , is the subject of limited analyses in advanced texts and periodicals on machine design . in the present invention , however , properties of motion have been found in the specialized form of this type of linkage not heretofore recognized . it has been further found that this motion can be applied to a mass in a practicable manner for different types of engine balancing . in particular , this invention provides a method of obtaining a single motion having a combination of harmonics of acceleration which are of the frequency and magnitude desired for engine balancing , without the introduction of undesired harmonics in any significant magnitude . this is accomplished by a proper choice of length of parts , location of pivot points , and size and location of balance weight . the balance rocker fixed bearing 17 is so located that when the balance weight 16 is midway in its arc of travel , a plane containing the axes of bearings 17 and 18 makes approximately a right angle with a plane containing the axes of bearings 13 and 18 . the latter plane approximates a horizontal plane in fig1 while the former approximates a vertical plane . true horizontal and vertical planes are defined for reference by the three intersecting dashed lines , which are the three principal axes of a dimetric projection . now assume , for example , in fig1 that the center of gravity of the balance weight 16 is also in the plane containing the axes of bearings 17 and 18 . when the balance crank 11 is rotated at a uniform angular velocity , as in an engine at constant speed , three important harmonics of acceleration will be produced at the axis of the balance weight bearing 18 . in a horizontal plane there will be a primary harmonic of acceleration having a frequency equal to the speed of rotation of the crank 11 . this acceleration applied to the balance weight 16 will result in a primary harmonic shaking force in a horizontal plane having a peak value which is defined by the expression where g is the acceleration of gravity and ω is the speed of rotation of the crank 11 in radians per unit time . also in a horizontal plane , there will be a secondary harmonic of acceleration having a frequency equal to twice the speed of rotation of the crank 11 . this acceleration applied to the balance weight 16 will result in a secondary harmonic shaking force in a horizontal plane having an approximate peak value of the expressions for primary and secondary shaking forces given above for a horizontal plane are similar to the usual expressions for shaking forces produced by reciprocating parts in a cylinder . it has also been found in the case of the balance weight 16 motion that there is a sizable secondary harmonic of acceleration in a vertical plane . its frequency is also equal to twice the speed of rotation of the crank 11 . the value of this secondary harmonic is such that when applied to the balance weight 16 there will result a secondary harmonic shaking force in a vertical plane which has an approximate peak value of analysis further shows for the ratios of lengths of parts chosen that the secondary harmonic of acceleration is the only harmonic of any consequence in a vertical plane . there does appear to be a trace of a primary harmonic , but contrary to the usual case with harmonics , its magnitude is extremely small relative to that of the secondary . the sizable harmonics in the acceleration of the balance weight 16 have been found to be the primary and secondary in a horizontal plane and the secondary in a vertical plane . these harmonics have frequencies and magnitudes which make them useful in engine balancing . in addition to these harmonics , there are other inherent harmonics of higher frequency in both the horizontal and the vertical planes which are not useful . these harmonics are small enough in magnitude , however , to be of little or no consequence . the largest of these higher harmonics is a tertiary or third - order harmonic of acceleration having a frequency equal to three times the speed of rotation of the balance crank 11 . there are tertiary harmonics of acceleration in both the horizontal and vertical planes , and they have approximately the same magnitude . the combined effect of these higher harmonics applied to the balance weight 16 is to produce a tertiary rotating force of approximately constant magnitude , which rotates in direction at three times the speed of rotation of the balance crank 11 . the approximate magnitude of this force has been found to be it has been found possible to choose lengths r b , l b and r such that ( r b / l b ) and ( r b / r ) are both small fractional values . in an application to a v - type engine , for example , the ratio ( r b / l b ) is in the approximate range from ( 1 / 7 ) to ( 1 / 10 ), while the ratio ( r b / r ) is in the approximate range from ( 3 / 7 ) to ( 3 / 10 ). the product of these fractional values in the expression for tertiary rotating force results in a very small magnitude for that force . for convenience in further discussion , an assembled unit such as in fig1 consisting of the balance rod 14 and balance rocker 15 , including the associated bearings and balance weight 16 , will be referred to as a balancing unit . the balancing unit of this invention , such as described above , is applicable to several types of engines which heretofore have been recognized as having considerable amounts of primary and secondary inertia unbalance . in general , such engines have four cylinders or less . as will be pointed out hereafter , the balancing unit of the present invention makes possible the design of certain v - type , four - cylinder engines having a combination of desirable characteristics not heretofore obtainable . to provide accurate conformity of the counterbalancing effect with the shaking forces of the engine , it is preferred that the angular position of the balance crank 11 be displaced from the dead center positions of the engine by an angle differing slightly from the conventional 90 ° angle . more specifically , the present invention comprehends displacing the angular position from the 90 ° relationship by a small phase angle of approximately ( 3r b 2 / 81 b r ) radians . where the balancing unit lengths have a relationship as discussed above , i . e ., r is approximately 21 / 3 to 31 / 3 times r b and l b is approximately 7 to 10 times r b , the small phase angle is approximately 1 °. more specifically , in referring to fig1 when the crank 11 is rotated 90 ° from the illustrated position so that the crank 11 is in the same plane as rod 14 , the balance unit may be described as being in a dead center position . thus , a pair of dead center positions is obtained at opposite sides of the upright position as shown in fig1 . the harmonic shaking forces developed by the engine are at a maximum when the crank or piston thereof is at either one of its dead center corresponding positions . however , the counterbalancing forces of the crank and rocker mechanism are not at a maximum when the crank 11 is at the above discussed dead center positions thereof . the maximum counterbalancing forces obtain at the above discussed small angular displacement from the dead center positions . in the illustrated embodiment of fig1 the displacement is counterclockwise therefrom . it has been found that a 1 ° error in the accurate positioning of the maximum counterbalancing forces relative to the corresponding angular disposition of the maximum engine shaking forces produces an unbalance of approximately 13 / 4 % of the primary harmonic forces . thus , the unbalanced force appears as an unbalanced force 90 ° from the angular position of the balancing of the engine shaking forces . by angularly displacing the counterbalancing unit as discussed above , this appreciable unbalance is effectively eliminated . illustrated in fig2 and 3 is an example of such a new v - type , four - cylinder engine having an embodiment of the balancing unit of the present invention as an integral and essential part . these views show in schematic form the principal moving components of the engine . this engine , for example , operates on the wellknown four - stroke cycle . power is developed in the usual manner for this cycle and transmitted through the four reciprocating pistons 19 , to the four connecting rods 20 , and to the four rotating power cranks ( or power crank pins ) 21 of the crankshaft 12 . the two power cranks 21 serving the pair of pistons 19 in each bank of cylinders are , for example , indexed 180 ° from each other . also , for example , the power cranks at either end of the engine pg , 14 for each of the two banks reach top or bottom center positions simultaneously . these conditions insure even firing intervals for any chosen angle 2α between cylinder banks . for the engine shown in fig2 and 3 , the angle 2α between banks has been chosen , for example , to be ninety degrees . a greater or lesser angle 2α could be used satisfactorily , however . also , uneven firing intervals could be obtained by changing the angular index of the cranks 21 , if such were desired . two balancing units , which for convenience are somewhat similar to that in fig1 are used to balance the engine . the balancing units are driven from the crankshaft 12 by the two balance cranks ( or balance crankpins ) 11 . the two balance cranks 11 are so indexed angularly between power cranks 21 at each end of the crankshaft 12 , that the two balance weights 16 move inward and outward essentially in phase with each other and approximately ninety degrees out of phase with the reciprocating motion of the pistons 19 . the balance rocker fixed bearings 17 are so placed , for example , that the center of gravity of each balance weight 16 makes approximately equal arcs of travel on either side of a vertical plane ( for an upright engine ) through the axis of its balance rocker fixed bearing 17 . also , for example , the same centers of gravity at any instant are in essentially the same horizontal plane ( for an upright engine ). it is the simplest choice but not necessary at any time that such plane contain the axis of the crankshaft 12 . in the engine of fig2 and 3 , for example , such horizontal plane is kept substantially below the axis of the crankshaft 12 . design for this condition causes the balancing units in fig2 to differ slightly in orientation and appearance from that shown in fig1 . as in fig1 a true horizontal plane is defined for reference in fig2 by intersecting dashed lines . the crankshaft 12 is shown supported by three crankshaft main bearings 13 . however , a different number of main bearings could be used , if preferred . the crankshaft 12 is also shown with two crankshaft counterweights 22 near the ends of the crankshaft . these counterweights 22 are positioned to balance the crankshaft 12 and associated rotating masses in addition to balancing a rotating primary couple produced by reciprocating masses . an additional pair of counterweights could also be placed on the inner arms of the two balance cranks 11 . other arrangements of counterweights are also possible . the static structure of the engine , now shown in the schematics of fig2 and 3 , consists of the usual cylinders surrounding the pistons 19 , crankcase with bulkheads supporting the main bearings 13 , cylinder head , and other common attachments . inasmuch as such a static structure is largely conventional it has been omitted in fig2 so that the novel features of the invention can be emphasized . the only important departure required from the usual v - type engine block construction is an extension of portions of the side walls to provide enclosures for the two balancing units and support for the pins which act as journals for the balance rocker fixed bearings 17 . the two required such enclosures may also be considered as direct extensions of the crankcase bulkheads supporting the crankshaft main bearings 13 . through proper selection of dimensions , the balancing units do not increase the overall width of the v - type engine . referring now , for example , to the dimensions indicated in fig2 the peak magnitude of the primary harmonic shaking force developed in one cylinder is indicated by the expression ( w a / g ) r a ω 2 where w a is the weight of the reciprocating parts for one cylinder ( mostly that of the piston ), r a is the radius of the crank circle of the power cranks 21 , ω is the speed of rotation of the crankshaft 12 in radians per unit time , and g is the acceleration of gravity . the shaking force attains its peak magnitude at top and bottom center of the crank 21 . the magnitude of the force is zero when the crank is 90 ° from top or bottom center . considering now one bank of two cylinders operating on cranks , for example , 180 ° from each other , the two primary shaking forces produce a primary rocking couple in the plane of the two cylinders . the peak magnitude of this couple is ( w a / g ) r a ω 2 a a where a a is the distance between center lines of the two cylinder bores , as indicated in fig2 . if two banks having two cylinders each are considered , the magnitudes of the primary rocking couples for the two banks are seen to be equal . the couples are also in phase , but they are in different planes , separated by the angle 2α between the two banks of cylinders . when the two couples are combined , the peak magnitude of the resultant couple is 2 ( w a / g ) r a ω 2 a a cos α . this resultant couple occurs in a plane midway between the cylinder banks , which is a vertical plane for an upright engine . referring to a description of the balancing unit of the present invention such as shown in fig1 the peak magnitude of the primary harmonic shaking force in the horizontal plane was shown in equation ( 1 ) to be ( w b / g ) r b ω 2 . referring next to the description of the incorporation of the two balancing units in the engine of fig2 and 3 , the design and orientation of the units is such that the two primary shaking forces , described as in a horizontal plane in fig1 produce a primary rocking couple in a horizontal plane for an upright engine . the peak magnitude of this couple is ( w b / g ) r b ω 2 b where b is the distance parallel to the crankshaft 12 between the centroids of the distributed unbalance forces developed by each of the two balance weights 16 in rotation about their respective balance rocker fixed bearings 17 . it will be noted in fig2 that b is greater than a a , for example , the latter being shown to be the same , for example , as the distance between center lines of the two cylinders in one bank . the distance b is larger than a a because the center of gravity of each balance weight 16 is offset from the center line of the balance rod 14 . this is a means of attaining a minimum weight for each of the two balance weights 16 while utilizing a simple construction . it is entirely possible , of course , to use other constructions for the balancing units and different relations between b and a a . the primary rocking couple ( w b / g ) r b ω 2 b , developed by the balancing units , is in a horizontal plane , while the primary rocking couple 2 ( w a / g ) r a ω 2 a a cos α , developed by the reciprocating parts in the cylinders , is in a vertical plane . inasmuch as one of the planes is 90 degrees from the other and one of the rocking couples is 90 degrees from the other in phase , the two rocking couples combine to give a rotating couple , which rotates with the crankshaft . the rotating couple will have a constant magnitude if the two component rocking couples are equal in peak magnitude . this occurs when this equation can be simplified to give the necessary weight of a balance weight 16 at the axis of the balance weight bearing 18 , or at an equivalent radius r from the axis of the balance rocker fixed bearing 17 , as now , there are three primary rotating couples which must be considered : ( 1 ) the constant rotating couple obtained as above , ( 2 ) the rotating couple unbalance inherent in the uncounterweighted crankshaft itself , and ( 3 ) the rotating couple produced by the ends of the connecting rods 20 and balance rods 14 rotating with the crankshaft . these all add to produce one large couple , which rotates with the crankshaft . this couple can be directly balanced by crankshaft counterweights , such as the counterweights 22 . calculations show that the necessary balance can be obtained with counterweights 22 of reasonable size . these counterweights 22 are similar in appearance and function to the end counterweights used on many v - 8 automotive engines . intermediate counterweights could also be added , with a resulting reduction in the size of the counterweights shown in fig2 and 3 . in addition to primary harmonic shaking forces , reciprocating parts in engine cylinders develop secondary harmonic shaking forces . these forces alternate through two cycles in one revolution of the crankshaft . the peak magnitude of the secondary harmonic shaking force developed in one cylinder is approximately where r a is the radius of the crank circle of the power cranks 21 and l a is the length of the connecting rod 20 . as in an in - line four - cylinder engine , the secondary forces in a v - 4 engine are in phase and thus reach their peak values simultaneously . while the forces in the two banks are in phase , they are in different planes , which are separated by the angle 2α between the two banks of cylinders . the resultant in a vertical plane through the axis of the crankshaft 12 for an upright v - 4 engine is a secondary shaking force having a peak magnitude of approximately in addition , because the pairs of forces in the two banks , or planes , are not directly opposite each other , a secondary rocking couple is produced in the horizontal plane . the peak magnitude of this couple is approximately where a h is the offset , or stagger , parallel to the crankshaft , between the two banks of cylinders , as indicated in fig2 . referring again to the description of the balancing unit of the present invention , such as shown in fig1 the peak magnitude of the secondary harmonic shaking force in the horizontal plane was shown in equation ( 2 ) to be approximately referring next again to the description of the incorporation of the two balancing units in the engine of fig2 and 3 , the design and orientation of the units is such that the two secondary shaking forces , described as in a horizontal plane in fig1 produce a secondary rocking couple in a horizontal plane for an upright engine . the peak magnitude of this couple is approximately the secondary rocking couple ( r b / l b )( w b / g ) r b ω 2 b , developed by the balancing units , is in a horizontal plane , and the secondary rocking couple 2 ( r a / l a ) ( w a / g ) r a ω 2 a h sin α , developed by the reciprocating parts in the cylinders , is also in a horizontal plane . furthermore , the phase relations of these two couples make them directly opposed to each other . for overall balance then , the two couples must be equal in peak magnitude . this occurs when ( r . sub . b / l . sub . b )( 2w . sub . a / g )( r . sub . a / r . sub . b )( a . sub . a / b ) r . sub . b ω . sup . 2 b cos α = 2 ( r . sub . a / l . sub . a )( w . sub . a / g ) r . sub . a ω . sup . 2 a . sub . h sin α this equation can be simplified to give the necessary length of the balance rod 14 approximately as it was shown in the analysis initial to the above that the secondary harmonic shaking forces from the individual cylinders also give a resultant secondary shaking force in the vertical plane of the engine . the peak magnitude of this secondary shaking force in a vertical plane through the axis of the crankshaft 12 for an upright engine was shown to be approximately referring again to the description of the balancing units of the present invention , such as shown in fig1 the peak magnitude of the secondary harmonic shaking force in a vertical plane was shown in equation ( 3 ) to be approximately referring next again to the description of the incorporation of the two balancing units in the engine of fig2 and 3 , the design and orientation of the units is such that the two secondary shaking forces , described , as in a vertical plane in fig1 add to produce a resultant secondary shaking force in a vertical plane for an upright engine . the peak magnitude of this force is approximately the secondary shaking force 2 ( r b / r )( w b / g ) r b ω 2 , developed by the balancing units , is in a vertical plane , and the secondary shaking force 4 ( r a / l a ) ( w a / g ) r a ω 2 cos α , developed by the reciprocating parts in the cylinders , is also in a vertical plane . also , for an engine which is symmetrical in design , for example , these two forces can be considered as coincident , since they are both fixed in point of application at all times at the geometric center of the engine . furthermore , the phase relations of these two forces make them directly opposed to each other . for overall balance then , the two forces must be equal in peak magnitude . this occurs when 2 ( r . sub . b / r ) ( 2w . sub . a / g ) ( r . sub . a / r . sub . b ) ( a . sub . a / b ) r . sub . b ω . sup . 2 cos α = 4 ( r . sub . a / l . sub . a ) ( w . sub . a / g ) r . sub . a ω . sup . 2 cos α this equation can be simplified to give the necessary length of the balance rocker 15 approximately as it will be noted throughout the preceding description that all expressions for secondary forces and couples and for the lengths 1 b and r , derived from the motion of the reciprocating parts in a cylinder and from the motion of an oscillating balance weight 16 , are described as approximate . this has been done because the simplest expressions for motion have been chosen . in the case of the motion of the reciprocating parts in a cylinder , a more accurate expression , involving a series of terms , is given in textbooks on the mechanics of engine design . in the case of the motion of an oscillating balance weight 16 , more accurate expressions , also involving a series of terms , have been derived , in connection with this invention . the simple expressions , however , are much to be preferred in a description of principles , and they also have sufficient accuracy for engine layout studies . more precise expressions will change the length l b in equation ( ii ) and the length r in equation ( iii ) around 1 or 2 percent . equation ( i ) is given in its precise form . as previously noted , it has been found that a balancing unit has a very small unbalanced tertiary rotating force . the approximate magnitude of this force , as given in the discussion of fig1 is it has been further found that for two balancing units arranged as in fig2 and 3 , the two tertiary forces are phased so that they are always in the same direction . for this reason , these forces add , and the approximate magnitude of the unbalanced tertiary rotating force for the whole engine is with substitutions from equations ( i ), ( ii ), and ( iii ), this expression for unbalanced tertiary rotating force can be given in more convenient terms as 9 / 2 ( r . sub . a / l . sub . a ). sup . 2 ( a . sub . h / a . sub . a ) ( w . sub . a / g ) r . sub . a ω . sup . 2 sin α this is relatively a very small unbalanced force ; its magnatude is approximately one - twelfth of the peak magnitude of the secondary shaking force developd at the same engine speed by a conventional in - line , four - cylinder engine having the same cylinder dimensions . the unbalanced tertiary rotating force given above is equivalent in magnitude , but not in speed of rotation , to the force developed at the same engine speed by a static unbalance of a design study has shown that this approaches in magnitude the total static unbalance permitted in present day automotive engines , in which the various moving parts are balanced by drilling or milling operations . a tertiary force also has much lower transmissibility through engine mounts . it may be concluded , therefore , that the tertiary rotating force is small enough to be ignored . mathematical analysis has shown that there is also a series of other small unbalanced harmonic forces inherent in the balancing units and also in the reciprocating parts of the engine . the small harmonic forces from both these sources are similar in type and comparable in magnitude to the unbalanced higher harmonics inherent in more conventional piston - type automotive engines . in the case of the subject v - 4 engine , mathematical analysis has further shown that these other small unbalanced harmonic forces are considerably smaller in magnitude than the unbalanced tertiary force , which itself has been found small enough to be ignored . as stated above , the inertia torques from the individual cylinders of a conventional four - cylinder engine add to each other . this produces a torque variation which is not desirable . one of the advantages in the use of the balancing units of the present invention , as illustrated by the embodiment of fig2 and 3 , is that the characteristic inertia torque variation for a four - cylinder engine is reduced in magnitude . this is due to the fact that the inertia torque produced by the balancing units directly opposes the inertia torque produced by the reciprocating parts in the engine cylinders . a design analysis shows that the use of the balancing units reduces the characteristic inertia torque variation for a four - cylinder engine by approximately 25 to 40 percent , depending upon detail design . while the inertia torque variation is reduced , the total effective rotating inertia of the engine is increased . the latter effect is also desirable , since it reduces the amount of rotating inertia which must be built into the engine flywheel . it is therefore apparent that when the weights and lengths of the parts of the balancing units of the present invention are chosen in accordance with equations ( i ), ( ii ), and ( iii ), using the more accurate forms for these equations , an engine such as illustrated in fig2 and 3 , will be essentially balanced as to primary and secondary forces and couples . all other forces and couples are insignificant in magnitude and for all practical purposes may be ignored . in addition , the inertia torque variation is appreciably reduced with the use of such balancing units . equations ( i ), ( ii ), and ( iii ) specify certain relations between weights and lengths of various parts for engine balance . there remains the practical consideration of the net effect of these specific relations on engine weight , overall dimensions , and complexity . in design studies it has been found that the above required relations for balance , in addition to other important characteristics required in a modern engine , can all be obtained in a simple , compact , and rigid engine structure . good results are obtained when the angle 2α between cylinder banks is in the approximate range of 70 to 90 degrees . the use of the balancing units does not increase the overall width of such a v - type engine . the following values are typical of those which may be used in the construction of an embodiment of an automobile engine patterned after the schematics of fig2 and 3 , and using the more accurate forms of the simplified equations ( i ), ( ii ), and ( iii ): the bore / stroke ratio represented by the above is 1 . 36 . this ratio is established by considerations of structural rigidity and compactness as about the optimum for this type of engine . such ratio also happens to be very favorable for a high - speed , high - efficiency engine . the crankshaft 12 of this engine is characterized by unusually large overlaps between all adjacent journals . this permits a short and very rigid crankshaft structure . the angle 2α between cylinder banks chosen for the above example is ninety degrees . this is an angle for which manufacturing facilities in the industry are presently adapted . the use of this angle 2α in the above was made possible by using balancing units in which the axes of the balance weight bearings 18 are constrained to move in paths substantially below a horizontal plane ( for an upright engine ) containing the axis of the crankshaft 12 . such position of the balance weight bearings 18 is obtained without departing from the previously described correct motion for the balance weights 16 by adjusting the angular position of the balance cranks 11 on the crankshaft 12 . additional benefits obtained with this construction and bank angle 2α are lighter balance weights 16 , reduced bearing loads , and a strengthened crankshaft 12 . several variations of the crank arrangement illustrated in fig2 and 3 can be made while still using the balancing principle described in connection therewith . identical results are obtained when either bank of cylinders is staggered ahead of the other . an alternate embodiment would simply be a mirror image of the illustration of fig2 and 3 . another possible variation may be achieved by interchanging each balance crank 11 with the endmost power crank 21 adjacent to it without changing the angular positions of the balance cranks or power cranks . this change results in a new configuration of the crankshaft in addition to changes in the geometry of the balancing unit to reestablish balance . identical results in this case also are obtained when either bank is staggered ahead of the other . the arrangement illustrated in fig2 and 3 or its mirror image is preferred because of its simpler and more rigid crankshaft structure . illustrated in fig4 and 5 is another form of v - 4 engine . this embodiment may use balancing units of the type already described . this engine also has essentially balanced primary and secondary forces and couples . the balancing units , however , increase rather than decrease the variation of inertia torque caused by reciprocating parts in the cylinders . this is a disadvantage of the engine as compared with the embodiment of fig2 and 3 , which is the preferred embodiment of an engine of this type . the embodiment of fig4 and 5 , however , can utilize a smaller bank angle 2α . also , no crankshaft counterweights , such as the counterweights 22 in fig2 and 3 , are required for overall balance . for this reason , no crankshaft counterweights are shown . however , counterweights can be added at any of several points along the crankshaft , as done with conventional engines , to reduce main bearing loads and structural bending forces . as with the engine of fig2 and 3 , it is not necessary at any time that the horizontal plane containing the axes of the bearings 18 also contain the axis of the crankshaft 12 . a variation similar to that for the engine of fig2 and 3 can be made by staggering either cylinder bank ahead of the other . also , as with the engine of fig2 and 3 , the balance cranks 11 and endmost power cranks 21 can be interchanged to give an optional balanced arrangement . referring now to fig6 there is here illustrated an in - line , four - cylinder engine with balancing units similar to those previously described . this engine is a conventional four - cylinder engine except for the provision of the balancing units of the present invention . in the conventional engine there is a large secondary shaking force in the vertical plane , or the plane of the cylinders , representing combined secondary forces from the four cylinders . in the application of balancing units , the balance weights 16 and associated linkage are designed to produce a secondary shaking force in the vertical plane , having a peak magnitude equal to that of the combined secondary forces produced in the engine cylinders . the two sets of forces are always directly opposed in direction so that they balance each other . also , the forces are distributed so that no couples are produced . in this application of the balancing units , only the vertical secondary forces of the balancing units are utilized . the effective masses of the two balance weights 16 are directly opposite each other , so that the primary and secondary forces in the horizontal plane from the two balance weights 16 oppose and balance each other , with no resulting couples . no crankshaft counterweights are required for overall balance . for this reason , no crankshaft counterweights are shown . however , counterweights can be added at any of several points along the crankshaft , as done with conventional engines , for the purpose of reducing main bearing loads and structural bending forces . several variations of the crank arrangement illustrated in fig6 can be made while still using the balancing principle described in connection therewith . identical results are obtained when the two balance cranks 11 are made to occupy positions approximately 180 degrees from the position shown in fig6 . the same results in most respects are also obtained when the two balance cranks 11 are made to occupy positions ninety degrees clockwise or ninety degrees counterclockwise from the positions shown in fig6 . in these two possible ninety - degree positions of the balance cranks the two balance rocker fixed bearings 17 must also be made to occupy fixed positions 180 degrees from the fixed positions shown in fig6 . however , the arrangement of balance cranks 11 shown in fig6 or the first variation described above , is preferred , because the balance weights 16 in such case reduce the characteristic variation of the inertia torque of the four - cylinder engine . in the case of each crank arrangement described above , it is not necessary at any time that the horizontal plane containing the axes of the bearings 18 contain also the axis of the crankshaft 12 . fig7 illustrates an in - line , two - cylinder engine with balancing units of the present invention , such as those previously described . the conventional in - line , two - cylinder engine is not favored by many manufacturers because of the unbalance characteristics of this type of engine . there is a large primary shaking force in the vertical plane , or in the plane of the cylinders , representing combined primary forces from the two cylinders . there is also a secondary shaking force in the vertical plane , also representing combined forces . in the application of an embodiment of the balancing unit of the present invention , the balance weights 16 are designed to produce a primary shaking force in a horizontal plane having a magnitude equal to that of the primary shaking force produced in the engine cylinders in the vertical plane . these two forces combine to produce a force of constant magnitude which rotates in the same direction as the crankshaft . this rotating force can then be balanced by the two crankshaft counterweights 22 shown . as in the case of the v - 4 engine , the length r of the balance rocker can be so chosen that the secondary shaking force produced by the balance weights 16 balances the secondary shaking force produced in the engine cylinders . in this application of the balancing units , only the horizontal primary forces and the vertical secondary forces are utilized . the effective masses of the two balance weights 16 are directly opposite each other so that the secondary forces in the horizontal plane from the two balance weights oppose and balance each other , with no resulting couple . referring now to fig8 there is schematically illustrated a balancing unit having a structure which differs in some respects from that shown in fig1 but which is designed for the same end results . the structural differences adapt the design for use with small industrial engines . like reference numerals are used in fig8 to those in fig1 for corresponding components . the principal difference between the structure shown in fig1 and that shown in fig8 is that balance rod 14 of fig1 is eliminated in the construction of fig8 . instead of such a balance rod , a spring means , schematically indicated as 15a is used . this spring means serves to keep balance weight 16 in physical contact with the driving crank pin of balance crank 11 when inertia tends to separate these components . spring means 15a may take a variety of forms . for example , a helical spring such as schematically illustrated may be used . in such an arrangement , one end would be anchored to the engine block or other suitable fixed or static member and the other end would suitably engage the balance weight 16 . alternately , the balance rocker 15 may be fixed at its upper end and may be provided with sufficient elasticity in its length to serve the dual function of a balance rocker and a spring . the crank pin of the balance crank 11 preferably should have a relatively large diametered cylindrical shape shown in fig8 ( rather than the smaller one schematically shown in fig1 ). as illustrated , the proportions of the balance crank 11 surface are such that they may be described as an eccentric . balance weight 16 is preferably formed as a cylindrical roller or with an arcuate surface in rolling contact with the crank pin . the distance 1 b in fig8 between the axes of the crank pin of the balance crank 11 and the balance weight 16 is the sum of the radii 1 1 and 1 2 ; 1 1 being the radius from the axis to the outer surface of the balance weight 16 , and 1 2 being the radius of the crank pin of the balance crank 11 from its axis to its surface . since the radii are constant , 1 b remains constant . thus , the balancing unit of fig8 can be designed with the same constant values of 1 b , r b and r as the balancing unit of fig1 . if , in addition , the distance between the fixed bearings 13 and 17 is made the same as in the embodiment of fig1 balance weight 16 of fig8 is capable of the same arcuate oscillating motion as the balance weight 16 of fig1 . consequently , the three algebraic expressions ( 1 ), ( 2 ) and ( 3 ), for harmonic shaking forces previously set forth for the balancing unit of fig1 may apply to the balancing unit of fig8 . the balancing unit of fig8 provides a design which is less costly to build than the balancing unit of fig1 . this is due primarily to the elimination of balance rod 14 in the design of fig8 . however , the design of fig8 is limited in application to engines which are considerably smaller than possible using the arrangement of fig1 because of limitations in dimensions and load capacities . for that reason , the design of fig8 would be more adapted for small and relatively inexpensive engines while the balancing unit of fig1 is more adapted for larger and more expensive engines . referring now to fig9 and 10 , there is illustrated an example of a small in - line , two - cylinder , four - cycle engine utilizing a variation of the design described in connection with fig8 . in - line , two - cylinder , four - cycle engines , although inexpensive , have not been popular because they have relatively poor balance . in the usual in - line , two - cylinder , four - cycle engines , primary shaking forces from each of the cylinders add to each other and result in a fairly large primary shaking force in the vertical plane or in the plane of the cylinders . similarly , the secondary shaking forces add to each other and result in a secondary shaking force in the same plane . as illustrated , a piston 19 is located in each of the cylinders 19a . the connecting rods 20 are each conventionally connected at one end to the piston and are connected at the other end to the power cranks 21 of the crankshaft 12 , which in turn is conventionally mounted in fixed bearings 13 . also secured to the crankshaft 12 , preferably adjacent the power cranks 21 , are the counterweights 22 . the balance crank 11 of the type described in connection with fig8 is secured to the crankshaft preferably intermediate counterweights 22 . the cylindrical surface of this balance crank 11 is engaged continuously by the balance weight 16 . this balance weight 16 in the preferred embodiment has an arcuate surface 16a adapted to engage the balance crank 11 . in addition , this balance weight may be provided with a lower extending flange 16b having a surface adapted to be engaged by spring 15a . spring 15a is a leaf spring having an intermediate section 15c adapted to receive and have secured thereto the flange 16b by suitable means . the ends of spring 15a may be secured to the static structure 19c by suitable means , such as by bolts 15d which extend through the spring into the structure 19c . the balance crank 11 is indexed 180 degrees from the two power cranks 21 . this means that the primary harmonic shaking force developed by the balance weight 16 is directly opposed throughout its cycle to the primary harmonic shaking forces developed in the two cylinders . it will be recalled that the peak value of the primary harmonic shaking force is determined by algebraic expression ( 1 ) supra . when the proper weight w b for balance weight 16 is selected , the primary force developed in the cylinders is balanced . furthermore , the secondary harmonic shaking force developed by the balance weight 16 in the same direction as the primary force developed by the balance weight 16 is directly opposed throughout its cycle to the secondary harmonic shaking forces developed in the two cylinders . this secondary harmonic shaking force is given by algebraic expression ( 2 ) supra . when the ratio ( r b / 1 b ) is made to equal the ratio ( r a ,/ 1 a ) the secondary force developed in the cylinders is balanced . in addition , all higher harmonic forces are also balanced . the symmetry of the two cylinders and the balancing unit further establishes that all couples balance . the arrangement illustrated provides a balance crank 11 which has a dual function , which includes the primary function of driving the balance weight 16 and secondary function of serving as an effective crankshaft counterweight . since the balance crank 11 also functions as a counterweight , it permits the reduction in size of conventional counterweights 22 . the total amount of crankshaft counterweighting required is that which will balance the crankshaft and the rotating ends of the connecting rods . the spring 15a in addition to its spring function also acts as a guide link . this spring 15a guides the balance weight 16 in a vertical straight line motion . the effective length r of the spring 15a , acting as a balance rocker in the embodiment of fig9 and 10 , is infinity , ( unlike r in fig8 which is a finite value ). with r equaling infinity , the peak value of the secondary harmonic shaking force developed in the horizontal direction is given by the algebraic expression ( 3 ) as : this is the desired value for utilization of the design of fig9 and 10 in a two - cylinder engine . while the embodiment of fig9 and 10 illustrates the use of a leaf spring , a coil spring may also be used . however , if a coil spring is used , suitable guidance for straight line motion of the balance weight 16 must be provided . the balancing unit in the engine illustrated in fig9 and 10 may also be used in one - cylinder engines . in such an arrangement the relevant parts would have a configuration as generally illustrated in fig9 . in such an arrangement , balance weight 16 would have to have a weight sufficient to balance the reciprocating parts in the cylinder . in a one - cylinder engine , however , the same symmetry of the balancing unit could not be obtained because of the longitudinal offset necessary between the power crank 21 and a balance crank 11 on the crank shaft . such lack of symmetry in a one - cylinder engine might result in a small unbalanced couple in such an engine . the important overall force balance , however , would be obtained . the present invention presents an important step forward in the art of internal - combustion engines of less than six cylinders by providing a combination of balance and other desirable features in such engines not heretofore achieved . while there has been described what is at present considered to be the preferred embodiments of this invention , it will be obvious to those skilled in the art that various changes and modifications may be made therein without departing from the invention , and it is therefore aimed in the appended claims to cover all such changes and modifications as fall within the true spirit and scope of the invention .