148 

 solved relatively to m r) and a lt 2 , a 3 , &c. are the roots of 



The author afterwards considers the equation 



CP 



I f(x, 0)<p(0)rf0 = F(ar), 



and the solution of a linear equation is at once made to depend upon 

 this as follows : viz. given for the determination of the function p x 

 the equation 



/(*, 0) ?(*) +/(*, 1) ^ + &c. = F(a). 



a, /3 being constants, and i//(0) a function of to be determined. It 

 is always permitted to assume this equation. 

 By this means, writing for shortness 



/!<*, 0) = /(*, 0) +/(*, 1)0+ &C., 



the equation becomes 



j: 



which is of the desired form. 



A solution which occurred to the author after the memoir was 

 drawn up, is as follows : viz. given, as before, the equation 



| 'x*0)f(*-HM =*<), 



then *(w V 1 being determined by the equation 



and putting, for abbreviation, 



e*W~ (.r- w v/~l) P c ^~)/(^ 0) d Q = f(x, wS~l), 



the equation 



