43 



The fraction (w) may be readily resolved into two fractions of 

 the form 



l+c',' 1+,-' 



where p +r = o q + s c- = — m^ Kc sin v 

 q + s = p + r c- zz — m^ K cos t 



m- ATcosct m- Ac sin or 

 p = - ____, q ^ ___ 



„ m'^Kcosrs m'^Kc sin nr 



r = , s = 



1— c^ ' 1— c» 



1+/ 



the fraction j-t^' may be considered as contained in 

 j^,i » Cj and C2 being arbitrary. 



whence - - '''^ ■ ^ ' + -^±^ — c. 

 Therefore, when n is odd and y — o 



d«M=d"3_ (—1) -Tc2 dv" + (—1) 2 9 c"-' d»" 

 when n is even and ► =r o 

 d"M =^"3 =(— l)^c, d i' + (— 1)> C dv" 

 These equations are the nlh particular fluxions of 

 M = c, cos v + C2 sin K + p cos c c + -f- sin c » — Cj 

 comparing the second fluxion of this equation with the giveii 

 equation 



— cs = G 

 Therefore substituting for p and q their values above given 



M _ Ci cos !> + Co sin V — i:::^z cos (c »— tt) + G 

 Case 2. When c = 1 



Equation (6) of this article gives 



H 2 



