Mr Sang on the Construction of Circular Towers. 261 



stacle, capable of resisting a horizontal thrust. Let that ho- 

 rizontal thrust be dh\ then have we, 



</«? — <//? cose +j»c?i sin « = . . (A.) 



dh^dp smi—pdi cosi—0 . . (B.) 



or, dw = d {p cos i) (A.) 



dh = d (p sin i) (B.) 



whence, integrating, 



TV sec i=p (C.) 



h=p sin i=w i2in I (D.) 



conclusions which might easily have been obtained without 

 the assistance of the integral calculus. 



In the second place, the stone GHIK may be supposed to 

 abut upon an inclined surface ; that is, the wall CDEF may 

 be supposed to be laid into a site ready prepared for it. The 

 retaining pressure is now not horizontal, but inclined to the 

 horizon at the angle i-\-^di. Let that pressure be dq \ then, 

 decomposed in the vertical direction upwards, it is dq . sin /; 

 and in a horizontal direction outwards, it is dq cos i : hence 

 the equations of equilibrium now are, 



dw = dq sin i+d (p cos . . . (E.) 

 =dq cos i-\- d (p sin i) . . . (F.) 



equations which are no longer integrable without a knowledge 

 of the nature of the curved surface, and of the law according 

 to which the thickness of the wall varies. 



As the theory in which 1 am at present engaged is essen- 

 tially that of retaining-walls, I may be allowed to pass be- 

 yond the lemmatic limits, and to exhibit the actual adaptation 

 of the strength to the strain. 



In the first case, when the retaining pressures are horizon- 

 tal, the investigation presents little or no difficulty. If t be 

 put for the thickness of the wall, and / for the length of the 

 curve, tdl will be proportional to the weight of the stone : 

 indeed, if we take one cubic foot of the stone as the unit of 

 weight, and consider only a length of one foot of wall, td! 

 will tiike the place of dw in equations (A.) and (B.). 



Hence we have, supposing ju to be proportional to t, that is, 

 p + nr. 



