LAPLACE. 507 



We might suppose that z^,^ is the coefficient of fr' in the ex- 

 pansion of a function of t and r ; then it would easily follow that 

 this function must be of the form 



^[t)^'^ (t) 

 nab \' 



Tti C 



\Tt T t J 



where </> (t) is an arbitrary function of t, and yjr (r) an arbitrary 

 function of r. 



Laplace, however, proceeds thus. He puts 



1 a h ^ 

 -c = 0, 



Tt T t 



and he calls this the equation generatrice of the given equation in 

 Finite Differences. He takes u to denote the function of t and t 

 which when expanded in powers of t and r has z^, y for the co- 



u 



efficient of fi^. Then in the expansion of -^-^ the coefficient of 



^V^willbe^,,,. 



Laplace then transforms —^ thus. By the equation generatrice 

 we have 



1 T 



' i-6 



T 



therefore, 



u \T h h) 



c + ah + a f h 



(I _ ,)^ 



Develope the second member according to powers of 5 ; 



thus 



u 



f~r 



,-'l(H'-''M"-'-^'-M"-^-] 



,ar-2 



, _ X (c + ah) a""'^ x(x — l), , ,,2 ^' 

 X l^'+-^"J + 1.2 (^ + ^^) 7l ^ 



T Vt y 



