130 Mr. Murphy on the 



This instance of the application of the present method is 

 sufficient to shew its nature, but as we may frequently facilitate 

 the operation by a slight previous transformation of the proposed 

 equation, I have added a few more examples. 



To find ss in the equation 



ass" + ss — b = 0, 



in this case, the rule may be directly applied; but the result 

 may be obtained, rather more simply, by putting %-b=x. 

 The equation becomes 



a(x + b) n + x = 0, 

 therefore by the rule 



#=coefficient of - in - 1. {1 + - .(x + b)"} 



X X 



= coefficient of - in - - (x + b)" + \ . J (* + by ~^.- s .(x + b) 3 " + &c. 



X X X X 



= -ab a + 2n.b-—\— =—= — .V"*.— + &c. 



2 1.2 3 



As another example, suppose a + ss=l. (ss) to find ss, 



or & — e".e z — 0. 



Hence by the rule 



ss = coefficient of- in - 1. (l-e". — ) 



1 e z e° a e s * e 3a e 3 " 



= coefficient of - in e" . - + — . — + — . -j + &c. 



ss ss 2 ss" 3 ss 



a qs 



— f a 4-X Z « So 4. 1 — c %a 4- &r 



-e-t-g.-.e + 3-j2 + <* c - 



