196 On Differential Equations. 



or 



V^"^ (2/0) — X (2/0) = z U+ b(p (2/0)] i^ (2/0) - (ae) 



and the solution of this equation, when practicable, will give 

 2/0 and Uq or/ (7/0) as a function of z. Suppose, then, that 

 from (ad) or (ae) we deduce 



2/0= ^W W 



then, after proper substitutions, (ab) becomes 



whose general term is 



dz'-^\{a^h^e{z)j dz j [r]'' " " ^^^ 



XI. The process by which the expansion (ag) has been 

 deduced resembles that by which the theorems of Laplace 

 and Lagrange may be obtained. But the theorem here 

 given is more general than either of those just mentioned, 

 both of which may be derived from the theorem (ag). 

 Thus, let 



a = \, 6 = 0, F{^y) = l, ^{y) = ^{y), | 



x(2/) = 0, ^(0) = V^(0) j 



then (ag) gives Laplace's theorem. 



XII. In addition to the conditions (ah) let the condition 



ylr(io) = iu ------- (ai) 



be satisfied. Then (ag) is equivalent to Lagrange's theorem.* 



* This Paper was finished at Brisbane on April 10, 1S6G. — J. C. 



