Mr Sang on the Construction of Circular Towers, 265 

 tern. The equation may be put in the form 



c sec i . I 

 rv-^''. e ^ 



and on making / zero, that is, on taking the state of matters 

 at the apex of the cone, 



c 



{w) = ^ 



and thus it appears, that, even at the apex of the cone, there 

 must be some extraneous weight. Substituting, we have, 



sec i . I 



w ~ iw) e " 

 sec i 



or, if we put e '* = E, 



and thus it appears, that the weights above two given points 

 D and E have their ratio measured by the intervening dis- 

 tance DE. 



Having ascertained the entire weight piled above a given 

 point D, we can proceed to inquire how the thickness of the 

 wall is to be regulated so that this arrangement may be 

 brought about. 



Differentiating, we have, 



- . V sec i •j^ ,, . 



dro = (fv) E dl; 



^ n 



but dw = r,2crtdlf taking the weight for the whole circuit 

 of the building, and hence 



(w) E = livrt = 2'!rtl sin 2, 



^ n 



(w) f! 



or, t = -- V^. -y 

 nsui2t I ' 



A glance at the above value will shew that the thickness / 

 has a minimum value. We have in fact 



ftsm2» /« 



