42 



stant, n Ueing odd and not less than 3. 



11 — 3 



d"u + d"-^u d y' - —m" (— 1 ) ~ Kc"-'- sin ar d p" (3) 



Let 2 be a function of u 



such that when v=o 

 d^u — d" z 

 Then equations (2) and (3) become 



J!— 2 



d" z+d^'-z dr — (—1 ) ~m- Kc"-'^ cos zndv" (4) 



d» 2 + d"-^z d /2= -(— 1 ) '^m^ Kc"-^ sin ^ d v" (5 ) 



Now when v—o 



the nth particular divided fluxion of '— . ,, 



when n is odd and not less than 3= dv^ d""" i+7v 



\n — 3 /» 71 — 3 7 71 



— ( — l)"^ *^ " '^'' ^"d when n is even = o 

 and the rath particular divided fluxion of 



n — 2 n — 2 n-j 



v" J when n is even and not less than 2 = ( — O's" c (/► C 



i+F? I- when n is odd = a 



Hence equations (4) and (5) are deduced from the integral 



(I+k') 2 = C1+C2 v+Cs y' — »'"• ATcim ,^.,3 + m' gco^ ^..^ ^^ 



' 1+;^ crT7^7)-(TR)- («'^ _ C^-' 



Where Ci, Cg, Cj are constant quantities, of which two are arbitrary 



with respect to the given equation. 



The former fraction = "' ~^' "*" " '■' — Cs and therefore maybe written 



