on 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, the whole decrement from the crown downwards is the sum or integral of all such partial decrements, and therefore the horizontal thrust transmitted to P is expressed by the symbol - w=fy8x.
But the resultant of these two pressures must be perpendicular to the joint of the arch-stones, or parallel to the line of the curve • wherefore ultimately we obtain, as the con- dition of equilibrium in such a structure, the equationSz(II - fy8z) = SxfySx Since the vertical pressure at P is so, while the horizontal strain is h, the intensity of the oblique strain at P must be ,/(w2 +M). Now, in passing to the proximate point Q, so becomes so + 8w, while it is reduced to it - Stoaz - so that the oblique strain at Q must be - J( (20 + 802 ÷ (it - Stutz)2 ), zu =1 - wherefore the strain at Q is „,/(w2 + h2), or exactly the same as that at P. This result might have been obtained from the consideration that the thrust upon the surface PQ is perpendicular to the oblique strain, and can tend neither to augment nor to diminish it. Hence, as a characteristic of this arrangement, we have the law that the tension across the joints of the arch-stones is the same all along, and therefore is equal to H, the horizontal tension at the crown of the arch.
From this it at once follows that if r be the radius of curvature at the point P, y being the vertical thickness of the mason-work there, H = ry, so that if R be the radius of curvature at the crown of the arch, and A the thickness there, the horizontal thrust there, or the strain transmitted along the arch-stones, is H = RAH, being measured in square units of surface ; hence also A : y :r :11, or the thickness at any place, is inversely proportional to the radius of curvature there.
When the form of the intrados is given, its curvature at any point is known, and from that the thickness of the stone-work and the shape of the extrados can be found. The most useful case of the converse problem is, again, that in which the extrados is a horizontal straight line.
Let OH, figure 6, be the horizontal