96 Mr murphy, on THE RESOLUTION OF 



A = <p{A), y = (j>'{A), 

 err = ^ + c.<P'{A)> 



c.y' = ^^^ + c,<p"(A) + c.,i>'{A), 



The general law of which equations is thus expressed : 



„ atic^nc.)" cb"'^"HA) 



^■"-•'y =^1.2...6xl.2...cxl.2...rfx&c.-^ ^^^' 

 h, e, (f, &c. being regulated by the two conditions before mentioned. 



From hence we obtain the complete integral of the proposed equa- 

 tion iir+x = </)(?<,); for since 



f(z) = A + Bs + Cz- + Ds' + &c. 

 .-. u. = yi + By' + cAByJ + c,{Byy + cABy')\ &c. 

 The arbitrary constant B is determined as usual by assigning a particular 

 value to X, as 



u, = A + B + c,B' + c,B' + CsJ?*, &c. 



by the reversion of which series B is found in a series arranged ac- 

 cording to the powers of Uo — A. 



Before proceeding to any particular applications of this general solu- 

 tion, a few observations will be useful, 



I. When (p{ti:,) is of the form u^ + const., then A becomes generally 

 infinite, and 7 becomes 1, the solution therefore fails in this case, but 

 more generally it may be remarked, that it also fails when the value 

 of A deduced from the equation A = (p{A) satisfies the equation (p'(A) = 1 ; 

 for this, by making 7 = 1, renders infinite the coefficients c,, c^, &c. Such 

 cases of failure will shortly be separately considered. 



