the zoom lens of the invention will be better understood from the following descriptions made with reference to the drawings . the zoom lens comprises , from front to rear , a 1st lens group i of positive power , a 2nd lens group ii of negative power , and a 3rd lens group iii of positive power and further includes a 4th lens group iv for forming an image of an object . letting fi , fii , fiii and fiv denote the focal lengths of the lens groups i to iv respectively , e &# 39 ; 1 the interval between the principal planes of the 1st and 2nd lens groups i and ii when in the shortest focal length position , e &# 39 ; 2 the interval between the principal planes of the 2nd and 3rd lens groups ii and iii , ξ , η and γ the amounts of movement of the 1st , 2nd and 3rd lens groups i , ii and iii by taking the positions of the 1st , 2nd and 3rd lens groups i , ii and iii when in the shortest focal length setting as respective start points with a position sign being given when measured along the optical axis from the front to rear , ξa and ηa the total movements of the 1st and 2nd lens groups i and ii from the shortest to the longest focal length position , z the zoom ratio , and ka (= ξa / η / a ) the ratio of the total movement of the 1st lens group i to that of the 2nd lens group ii , we have the amount of movement γ of the 3rd lens group iii as a solution of the following quadratic equation : ## equ1 ## in the present invention , to vary the focal length of the engtire lens system , the 1st and 2nd lens groups i and ii are made to be axially movable in opposite directions to each other . in this case , as the 1st lens group i is pulled out forward by the actuator , the 2nd lens group ii has to move in driven connection thereto , so that when the amount of movement of the 2nd lens group ii is far larger than that of movement of the 1st lens group i , an unduly large stress is put on the operating mechanism . though this drawback can be overcome by using an alternate arrangement where the 2nd lens group ii is moved by the actuator , while the 1st lens group is made to follow it up , it is found that since the amount of movement ξ of the 1st lens group i is small , the advantage of the invention is reduced , being almost equal to that expected from the conventional type of zoom lens in which only one or the 2nd lens group ii is moved to effect a change in focal length . on the other hand , when the amount of movement ξ on the 1st lens group i is very large , the total length of the lens system when in telephoto settings becomes longer so that though the aberration correction becomes easy , the diameter of the front lens members must be increased to an objectionable point . in the present invention , a good result is obtained by satisfying the following conditions : where ka is the ratio of the total movement of the 1st lens group i to that of the 2nd lens group ii . also in the present invention , with the 1st and 2nd lens groups made to move on the common optical axis in opposite directions to each other rectilinearly so that the on - axis rays emerge out of the 3rd lens group iii almost in parallelism , it is made possible to eliminate variation with zooming of aberrations by simple techniques and also to minimize the zonal spherical aberration . it is to be noted that the 1st and 2nd lens groups i and ii may be moved non - rectilinearly . further , for a lens system in which the 3rd lens group iii takes the same position when in the shortest and longest focal length settings we have the following formula : where k is the ratio of the amount of movement ξ of the 1st lens group i to that of movement η of the 2nd lens group ii . at this time , the focal length fiii of the 3rd lens group iii is determined by : imparting of such perfect reciprocating motion to the 3rd lens group iii produces an advantage that , since the 3rd lens group iii lies at the same point in position for both of the shortest and longest focal lengths of the entire lens system and contributes equal aberrations to the system , all that is needed to be taken into account is the stability of aberration correction in the intermediate focal length region . it is therefore made possible to construct even the 3rd lens group in the form of a singlet lens . in order that the ratio k obtained by solving the above equations satisfies the before - mentioned condition ( 1 ), it is preferred to set forth the following conditions : when the lower limits of conditions ( 2 ) and ( 3 ) are exceeded , as the amount of movement of the 1st lens group i becomes relatively small to the amount of movement of the 2nd lens group ii , the refractive power of the 1st lens group i must be strengthened . if the number of lens elements constituting the 1st lens group is not increased , it becomes difficult to remove the variation of aberrations with focusing to shorter object distances . further , the amount of movement of the 2nd lens group ii increases . to avoid this , by strengthening the refractive power of the 2nd lens group ii , the refractive power of each of the other lens groups must be necessarily increased , thereby giving a disadvantage that the range of variation of spherical aberration with zooming is increased . on the other hand , when the upper limits of conditions ( 2 ) and ( 3 ), the amount of movement of the 1st lens group i becomes relatively large to the amount of movement of the 2nd lens group ii . this calls for an increase in the total length of the lens system when in the longest focal length positions and an increase in the diameter of the front lens members . in addition thereto , it becomes difficult to remove the positive distortion in the long focal length positions . therefore , the choice of a refractive power distribution satisfying conditions ( 2 ) and ( 3 ) makes it possible to reduce the increase of the petzval sum in a negative sense resulting from the limitation of the telephoto ratio to a minimum , for the total length of the lens system would be otherwise elongated in the long focal length positions , and makes it possible to eliminate variation of the spherical aberration resulting from the increase in the refractive power of each of the lens groups . moreover , since the total length of the lens system in the short focal length positions is short , the burden of carrying about the lens in attachment to the camera body can be considerably reduced . next , the numerical data for embodiments of the invention satisfying the above - cited conditions are given . the embodiments are zoom lenses having a focal length range of 103 to 197 mm with an f - number of 1 : 4 . 5 . in the following tables , ri is the radius of curvature of the i - th lens surface counting from the front , di is the i - th axial lens thickness or air separation counting from the front , and ni and νi are respectively the refractive index of abbe number for spectral d - line of the glass from which the i - th lens element counting from the front is made up . ______________________________________example 1f = 103 - 197 fno .= 1 : 4 . 5 2ω = 24 - 12______________________________________fi = 94 . 81r1 = 109 . 46 d1 = 2 . 30 n1 = 1 . 80518 ν1 = 25 . 4r2 = 59 . 49 d2 = 7 . 00 n2 = 1 . 61800 ν2 = 63 . 4i r3 =- 1037 . 84 d3 = 0 . 15r4 = 110 . 78 d4 = 4 . 00 n3 = 1 . 61800 ν3 = 63 . 4r5 =- 1888 . 22 d5 = variablefii =- 30 . 00r6 =- 256 . 99 d6 = 2 . 00 n4 = 1 . 77250 ν4 = 49 . 6r7 = 113 . 61 d7 = 5 . 54r8 =- 121 . 66 d8 = 3 . 90 n5 = 1 . 84666 ν5 = 23 . 9ii r9 =- 30 . 80 d9 = 1 . 50 n6 = 1 . 69680 ν6 = 55 . 5r10 = 234 . 21 d10 = 1 . 86r11 =- 55 . 78 d11 = 1 . 36 n7 = 1 . 69680 ν7 = 55 . 5r12 = 199 . 81 d12 = variablefiii = 73 . 71r13 = 126 . 42 d13 = 6 . 80 n8 = 1 . 65830 ν8 = 53 . 4r14 =- 28 . 53 d14 = 1 . 50 n9 = 1 . 84666 ν9 = 23 . 9iii r15 =- 55 . 96 d15 = variabler16 = 0 . 0 d16 = 1 . 50fiv = 135 . 63r17 = 49 . 80 d17 = 7 . 00 n10 = 1 . 58904 ν10 = 53 . 2r18 =- 87 . 10 d18 = 1 . 90 n11 = 1 . 84100 ν11 = 43 . 2r19 = 3540 . 44 d19 = 51 . 02iv r20 = 86 . 07 d20 = 4 . 00 n12 = 1 . 57845 ν12 = 41 . 5r21 =- 353 . 40 d21 = 5 . 98r22 =- 34 . 68 d22 = 2 . 00 n13 = 1 . 72000 ν13 = 46 . 0r23 =- 113 . 86d / f 103 . 0 150 . 35 197 . 0d5 1 . 97 17 . 97 26 . 21d12 17 . 80 9 . 66 1 . 64d15 4 . 00 1 . 47 4 . 00 ( fi + fii - e &# 39 ;. sub . 1 ). sup . 2 /( zfii . sup . 2 )= 1 . 53 -( fi + fii - e &# 39 ;. sub . 1 )( 1 - 1 / z )/ ηa = 1 . 5______________________________________ ______________________________________example 2f = 103 - 197 fno .= 1 : 4 . 5 2ω = 24 - 12______________________________________fi = 120 . 68r1 = 75 . 54 d1 = 2 . 20 n1 = 1 . 80518 ν1 = 25 . 4r2 = 48 . 97 d2 = 7 . 00 n2 = 1 . 53996 ν2 = 59 . 5i r3 =- 384 . 66 d3 = 0 . 15r4 = 157 . 01 d4 = 3 . 00 n3 = 1 . 51633 ν3 = 64 . 1r5 = 315 . 56 d5 = variablefii =- 36 . 0r6 =- 360 . 24 d6 = 1 . 60 n4 = 1 . 71285 ν4 = 43 . 2r7 = 44 . 79 d7 = 4 . 13ii r8 =- 42 . 08 d8 = 1 . 60 n5 = 1 . 74320 ν5 = 49 . 3r9 = 39 . 05 d9 = 5 . 00 n6 = 1 . 80518 ν6 = 25 . 4r10 =- 131 . 66 d10 = variablefiii = 81 . 35r11 = 168 . 47 d11 = 6 . 80 n7 = 1 . 62374 ν7 = 47 . 1r12 =- 30 . 21 d12 = 1 . 50 n8 = 1 . 80518 ν8 = 25 . 4iii r13 =- 54 . 76 d13 = variabler14 = 0 . 0 d14 = 1 . 50fiv = 144 . 04r15 = 53 . 59 d15 = 8 . 00 n9 = 1 . 63930 ν9 = 44 . 9r16 =- 81 . 31 d16 = 1 . 90 n10 = 1 . 84666 ν10 = 23 . 9r17 = 904 . 24 d17 = 70 . 65iv r18 =- 25 . 90 d18 = 1 . 78 n11 = 1 . 69350 ν11 = 53 . 2r19 =- 38 . 22 d19 = 0 . 15r20 = 61 . 74 d20 = 3 . 60 n12 = 1 . 67270 ν12 = 32 . 1r21 = 82 . 98d / f 103 . 0 150 . 35 197 . 0d5 1 . 63 25 . 14 37 . 26d10 16 . 56 8 . 58 0 . 72d13 4 . 00 1 . 52 4 . 00 ( fi + fii - e &# 39 ;. sub . 1 ). sup . 2 /( zfii . sup . 2 )= 2 . 25 ( fi + fii - e &# 39 ;. sub . 1 )( 1 - 1 / z )/ ηa = 2 . 25______________________________________ ______________________________________example 3f = 103 - 197 fno .= 1 : 4 . 5 2ω = 24 - 12______________________________________fi = 184 . 50r1 = 122 . 19 d1 = 2 . 50 n1 = 1 . 76182 ν1 = 26 . 6r2 = 61 . 12 d2 = 1 . 20i r3 = 62 . 09 d3 = 9 . 00 n2 = 1 . 60729 ν2 = 49 . 2r4 =- 335 . 00 d4 = variablefii =- 40 . 0r5 =- 112 . 40 d5 = 1 . 60 n3 = 1 . 71300 ν3 = 53 . 8r6 = 111 . 92 d6 = 4 . 70ii r7 =- 89 . 96 d7 = 1 . 60 n4 = 1 . 69680 ν4 = 55 . 5r8 = 32 . 55 d8 = 4 . 00 n5 = 1 . 80518 ν5 = 25 . 4r9 = 118 . 98 d9 = variablefiii = 73 . 81r10 = 184 . 12 d10 = 6 . 80 n6 = 1 . 58904 ν6 = 53 . 2r11 =- 28 . 94 d11 = 1 . 50 n7 = 1 . 80518 ν7 = 25 . 4iii r12 =- 44 . 98 d12 = variabler13 = 0 . 0 d13 = 1 . 50fiv = 139 . 58r14 = 41 . 40 d14 = 7 . 00 n8 = 1 . 61117 ν8 = 55 . 9r15 =- 226 . 83 d15 = 0 . 90r16 =- 108 . 81 d16 = 1 . 90 n9 = 1 . 84666 ν9 = 23 . 9r17 = 978 . 60 d17 = 65 . 28iv r18 =- 24 . 67 d18 = 1 . 78 n10 = 1 . 69680 ν10 = 55 . 5r19 = - 47 . 73 d19 = 0 . 15r20 = 91 . 58 d20 = 3 . 60 n11 = 1 . 59270 ν11 = 35 . 3r21 =- 14415 . 58d / f 103 . 0 150 . 35 197 . 0d4 1 . 43 44 . 10 66 . 09d9 12 . 60 7 . 17 1 . 83d12 4 . 00 2 . 32 4 . 00 ( fi + fii - e &# 39 ;. sub . 1 ). sup . 2 /( zfii . sup . 2 )= 6 . 0 ( fi + fii - e &# 39 ;. sub . 1 )( 1 - 1 / z )/ ηa = 6 . 0______________________________________