72 Mr. Gwyther 07t a 



And each of these is to be affected by c-s'mddddij) and 

 integrated between the Hmits of and 27r for (p and and 

 TT for a 



The integration with regard to <^ is easily done, and we 

 get 



^ depends onyg only 

 >/ depends ony^ and \pi, 

 and ^ is the subject of our further investigations. 



Resolving again parallel to the original axes we see 

 that to reproduce the original displacement at P we must 

 have 



hence we conclude ^3 = 0, ^^ = 0, and thus have further 

 simplified the formulae (i). 

 Then 



^= I / (cosacosO + sinasinf^cos^)'Vsin9^0^(^ 



O o 



X { [/2(x.r)ysmd - (pi{x.r)cosd)s'masm-({) 

 + (^(pi{x.r) + (f)6{x.r)y"^(^cosasmd - sinacos0cos(^)cos(/)}. 

 The integration with regard to (p can be completed thus 



/ (cosacos0 + sinasin0cos0)'"sinasin"0^0 



{Uliffl I ^ I 



cos"'acos"*0sina + ~-^ -^cos'"~"acos"'~-0sin^asin-O + etc. \ 

 2.4 J 



Let us separate in this the parts containing cos"'asina 

 cos"'~2asina, etc., and at the same time change the integral 

 from one with regard to to one with regard to r, by 

 writing 



;r + ^=fjcos6. 

 y = psinO 



We thus get as the coefficient of 7rcos'"asina/p"'+^ 



