S72 Mr MURPHY'8 THIRD MEMOIR ON THE 



41. General Calculus for the Cylinder with any law of force. 



Let «o, «i. a^.-Mn represent the total actions on the point A which 

 would be exerted by the successive portions P^Pi, PiPz of the shell, 

 if the density of each were unity; these quantities are given, since the 

 law of force is supposed known. 



Then aopo, a^p^, a^p^, &c. represent the actual forces on A. 



Again, the action of any portion as P„+4P„+5 on any point Q„ of 

 division in the axis, will be to the action of the similarly situated 

 portion P^P^ on the point A in the ratio of the corresponding densi- 

 ties, and in this case would be atpm+i. 



By this consideration the total actions on the points Qi, Q2...Q„ 

 are easily estimated, and equating each to zero, we get the following 



system of n equations, which serve to determine the ratios — , — , &c. viz. 



pa Pa 



aopo- Ctopi — aipz — Chps — asPi — dn-lpn-l — (tn-iPn = 0, 



aipo + aopi — aop-i — aipi — aipi —ctn-spn-i — an-ip„ = 0, 



Oipo + aipi + ttopi- ttops—aipi — «n-4/'n-l — «n-3/'« = 0, 



a^po + a^pi + aip2 + aopa— Oopi —ctt,-ipn-i — (i„^ip„ = 0. 



a„-2po + an-3pl +«„_4p2 + «n-5/03 + «»-6P4 — «2/0«_l — fl!l/0» = 0, 



ffn-lpO + fin-2Pl + «»-3p2 + «»-4j03 + «n-5/04 +«l/On-l " aoPn = 0. 



Comparing the first equation with the »*'', the second with the (w-l)"", 

 &c. it is obvious that po is involved in the same manner as /o„, p^ as 



Pn-l, &c. 



Hence, p„ = po, p„_i = pi, p„.i = p^, &c. 

 Form now two functions in the following manner: 

 a known function, M = «osin0 +«isin30 + a8sin50+ .,.«„_isin(2w-l)0, 

 an unknown, aS'„ = jOoCOSw0+|OiCOs(w-2)0 + ^sCOs(«-4)0+ 



