and their Index Symbol . 133 



where is a given function of x and y, in the form 



*=F(V).0 + F(V).O, (4) 



in which 



F(V) = {Av(V-l)..(V-«+l)+BV(V-l)..(V-/3 + l) + &c.}- 1 , 



and in which the value of the first term is perfectly definite, and 

 can be had at once by formula (2). It appears, then, that as far 

 as equations of this class are concerned, the number and character 

 of the arbitrary functions in a solution, which are due solely to the 

 second term, are unaffected by the number of independent 

 variables which the equation may contain, and are solely de- 

 pendent on its order. 



When the roots of the equation 



A.V(V-I). . .(V-a + l)+B.v(V-l). . .(V-/8 + l) + &c.=0 

 are all real and unequal, the arbitrary portion of the solution is 

 of the form 



u m + u n + u p + &c, 

 m, n, p, &c. being the values of the roots. 



When it contains a equal roots, whose common value is m, its 

 form is 



u m {\ogx + \ogy)*- l +v m .{\ogx + \ogy} a ~ 2 + &c. + u n + u p +kc., 

 where u m) v m , &c. are different arbitrary homogeneous functions 

 of the same degi'ee. Finally, when this equation contains pairs 

 of imaginary roots, the form of the arbitrary portion of the 

 solution is 



5. By a single very obvious reduction, similar to the first 

 which Legendre has employed {Memoir es de V Academie, 1787) 

 for the solution of the corresponding class of ordinary differen- 

 tial equations, we obtain at once, by the method of the last 

 article, the solution of the class of partial differential equations, 



{< 



d x z 

 m-\-\x)*j-^ +a{m + \x)*- 1 . {n + \y) 



1 



— ; — ~ (m + Xx) 



1.2 



d a z 

 ^ V' dx a -tdy 



dx*-idy 



+ &C.J- 



+ 



{< 



>& 



B \ (m + Xx^j- +£(m + }U)0 



tLcft 



r 



-1) 



{n + X ^dx^dy 

 dPz 



>=n, 



1 .2 



dPz ~\ 



.(»+x.)^-.(i.+>r)« asR5 i+fa.} 



+ &c. 



without the necessity of any further transformation. 



