es, and must therefore be parallel to the straight line drawn to touch the curve at P.
Hence, if the form of the inside of the arch, or the intrados as it is called, be prescribed, we can easily discover the law of the pressures at its various parts ; thus, to find the strain at the point Q, we have only there to apply a tangent to the curve and to draw kg parallel thereto ; kg represents the oblique strain at Q ; eq represents the whole weight from the crown A to Q, and therefore pq is proportional to the weight imposed upon the position PQ of the arc.
Using the language of trigonometry, the horizontal thrust is to the oblique strain at any part of the curve as radius is to the secant of the angle of inclination to the horizon; also the same horizontal thrust is to the weight of the superstructure as radius is to the tangent of the same inclination. And thus, if the intrados be a known curve, such as a circle, an ellipse, or a parabola, we are able without much trouble to compute, on this hypothesis, the load to be placed over each part.
If we use the method of rectangular co-ordinates placing x along OH and z vertically downwards, so that Pr may be the increment of x, irQ that of z, the tangent of the inclination at P is - and therefore if h stand for the horizontal strain, and so for the weight of the arch, we have - , "k to = azs while the oblique strain is h Rx2,• Also the ox?
change of weight from P to a proximate point Q is Szo = h8(Tx) .
Let RST be the outline which the mason-work would have if placed compactly over the arch-stones, in which case RST is called the extrados, then the weight supported at P is proportional to the surface ARSP, and the increment of the weight is proportional to PSTQ, hence if the weights and strains be measured in square units of the vertical section of the structure, and if y be put for PS, the thickness of the mason-work, we have - 8w = y8x, whence y= k - n ‘):2 When the curve APQ is given, the relations of z and of its differentials to x are known, and thus the configuration of the extrados can be traced, and we are able to arrange the load so as to keep all the strains in equilibrium.
But when the form of the extrados is prescribed and that of the intrados is to be discovered, we encounter very great difficulties. Seeing that our hypothesis is not admissible in practice, it is hardly worth while to engage in this inquiry ; it may suffice to take a single, and that the most interesting case.
If the whole space between the arch-stones and the roadway be filled up, the extrados becomes a straight line, and when this is horizontal we have y= z so that the form of the arch must be such as to satisfy the condition - z =, - that is to say, z must be a function of x such as to be proportional to its own second derivative or diferential coegicient. Now this character is distinctive of the catenarian functions, and therefore ultimately xi; A( e'Vi+e ), where A is AO, the thickness at the crown of the arch, and e the basis of the Napierian system of logarithms. In this case, since 8w= Ay, - so that the form of the areh and also its weight may readily be computed by help of a table of catenarian functions.
Let us now consider the case when the ends of the arch-stones are dressed continuously, while the imposed load is formed of stones having vertical faces. The weight of the column PSTQ resting on the oblique face PQ is prevented from sliding by a resistance on the vertical surface QT, which resistance goes to partly oppose the horizontal strain transmitted by the preceding arch-stone ; and thus the out-thrust of the arch, instead of being entirely resisted by the ultimate abutment, is spread over the whole depth of the structure. In this case the horizontal thrust against QT is to the weight of the column as Qr., the increment of 7, 13 to , the increment of x; wherefore, putting H for the horizontal thrust at the crown of the arch, and h for that part of it which comes down to P, the decrement of h. from P to Q is proportional to the rectaugle under PS and Qr, that is to say, Sk=y&.
Now, th