196 Royal Society : — 



tlie problem of determining the fimction (p which (/, F being given 

 functions, and the Umits a, /3 of the integration being also given) 

 satisfies the equation 



f /(^', 0) <p{x+e)cm=z'ex. 



J" 



He observes, that, unlike the methods followed in his former me- 

 moir, and the solutions there employed, which are quite rigorous, 

 the methods of the present memoir depend upon developments into 

 series, the strictness of which has been contested by some mathema- 

 ticians ; but that passing over these difficulties, he has solved the 

 famous problem, the solution of which has been vainly sought after 

 for the last two hundred years, because on the above-mentioned 

 equation depends the integration of the generally Imear equation of 

 any order whatever of two variables, and consequently the whole 

 Integral Calculus. The solution first obtained by the author, and 

 which he afterwards exhibits under a variety of different forms, is as 

 follows : — 



Theorem I. — The equation being given, 



I* /(^, 0) <p{x + d) c/0 = l^lj = F(a;), 



where f{x, 6) is a given function of x and d ; F„(x) is a given func- 

 tion of X such that the equation F^{x)=^cc cannot hold good for any 

 finite value of x ; Fj(a;) a given function of x containing all the 

 factors which render F{x) infinite, and the function F{x) being abso- 

 lutely arbitrary ; and a and /3 being given constants (independent 

 therefore of x and 6), the expression for <p x which satisfies the pre- 

 ceding equation is 



^^=/i(«i)Fi'K)' +fMF\i».)' -^^'•' 



where /.(a;) is determined by 



/(a;) = (x-a,) e**^"""^ C f(x, 6) e'"'^dd. 

 *(a,.) is a root of the equation 



=0 



^ a 



e^'^f{a„d)dd 



solved relatively to >w„ and a^, a^, a^, &c. are the roots of 

 F,(^) = 0. 

 The author afterwards considers the equation 



r/(a;,0)?i(0)rf0 = F(«), 



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

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



