44 



« = ^^ - '^^ Ti+o^ ^""^ 



let the term v be represented by 



(l+r)^ 



let also the part of m corresponding to v be called u\ 

 1. n being even and v = o 



^v = (—1) " G) i''^'" 



2. M being odd and v = o 



cin^ = (— 1)^~ (T-)qdv". Therefore d»ic =(— l)'^^!!") rfi-^-'cff 



all , 1 — 1 T 1 T \ 



or rf'w = —(—1) Urt" di—T,d\r) 



according as n is even or odd. 



These two equations are the particular divided fluxions of 



w= — - V sin 1/ + -^ f cos ► — i- sin D 



the latter term may be neglected, being contained in 



C2 sin V. Hence substituting for p and q 



u = Ci cos ►+ Cg sin v — i m^ Kv sin (i/— w) + G 



Case 3. When c = 1 and m- is a very small quantity, the following- 

 integration will be exact to the Hrst power of tm. 



By the same substitution as in the preceding case 



_ '^. +'■■' _ /"'+?'' _ c 



'^ ~ I +,■' (1 +,^ )» 3 



1 



now 





c,(l+,») 



nearly = T^"r+7^- 

 Hence z = 





-i + /i+£\, +i + /i+^^,^ nearly 



