98 Mr murphy, on THE RESOLUTION OF 



Examples : 



I. ih+, = a + bu, = <p (u^). 



Results 



u, = ^-^ + B.b' 



ll„ = r i T //. 



"°~ 1 - h 



II. Given m,+, = -^-^ — = <p{u,y 

 Immediately applying the general formulae above found, they give 



A =b + Jh' + d\ 7 = 1+ V (l + ?;) , 



1 Ci ^1.2 



'''- Uy{y- 1)' ''^-*7(y-l)' ''^"67(7^- >)' ''''■' 



and by substituting these values, we have 



{By'-y- {By'f 



"'^*'^+^'>''^2i7(7-l)^26V(7-l)(7^-l) 



(fi7-r.(7 + 5 ) ^ 



^26»7M7-l)(y- 1)(7'-1)' ■' 



and B js known by reverting the series 



B^ jgs 



""- ^-y = ^ + 2*7(7- 1) "^ 2iV(7-l)(y-l) "^ ^''' 



To facilitate the determination of B in every case of the general 

 solution, viz., u„ = A + By' + c,{By')- + ciBy'J, ^c, we may apply 

 the general theorem 



^ ^1 .2...(w- 1)(m„--^)'»(- 1)"-'" . c,''' c/« &c. 

 "-" 1.2. ..ft,, X1.2...6, X 1.2. ..ft. X &c. ' 



where the indeterminate indices are subject to the two conditions 



