Mr iSang on the Construction of Circular Towers. 263 



vertical planes passing through the axis AB, and inclined at 

 some small angle a. It is quite obvious, that the weights of 

 the parts under, consideration will be proportional to a, and 

 that, on this account, a must enter as a multiplier. 



Putting r for the radius c?D, the area of the upper bed of 

 the stone must be rat; and if n be the pressure on each 

 square foot of that area, nrat must take the place of p in our 

 former investigation, while ratdl takes place of dw. 



The tower being hollow, there is no substance to resist di- 

 rectly the advance of the stone towards de : that resistance is 

 afforded by the pressures against the ends of the stone, or, so 

 to speak, against the two bounding vertical planes. Put <r for 

 the end strain, and each square foot of D hi E, and the whole 

 end strain will be atdl, which, acting at an inclination a, is 

 decomposed into atdl sin a, in the direction dT> ; this ex- 

 pression atdl sin a, or, on account of the smallness of a, 

 6tdla takes the place of dh ; and we thus have 



w = I artdl = nart co^i .... (I.) 



A = / <r/fl?/ = nart^mi .... (K.) 



observing that ar^ = — -, we find 

 dl 



drv . ^ . 



w — n-— cos «, or (L.) 



dl sec i = n — (M.) 



w 



as before, so that if the structure be equilibrated, that is, if n 

 be constant, we must have as before 



n nep log w — I dl sec i ; 



but the thickness no longer follows the same law, for the re- 

 sisting surface, being extended in proportion to the radius r, 

 as well as to the thickness t, rt takes now the position which 

 t alone formerly occupied. To fix this point more firmly in 

 the mind, I may remark, that, in the case of a circular tower, 

 the thickness does not necessarily increase with the depth, 



VOL, XXIX. NO. LYIII.— OCTOBER 1840, B 



