64 M. C. Cellerier on the Distribution of 



F(w, v), we shall thus get 



where X corresponds to the segment ACBE, so that we have 



x= r<^ F(a)i F(M)= c- __vV±ZE? 0(l .)*). (it) 



Jo J V*2- M 2 U 



The other parts of the value of -\//(V), two of which are equal, 

 decompose into products. Putting 



<')-[« + &+f+jf]*V>.) . . (18) 



J (x) = 2x\ + lxfq + 2x>P. J 



<7 = I <p(v)dv= I cf>(u)du, g'= \ v 2 <f>(v)dv, 

 Jo Jo Jo 



we get 



*(*) = 



The first process which presents itself for finding (f>(x) con- 

 sists in assigning to it the form of a series a x x + a 2 a? + a z x z + ...; 

 we must then put 



r° <f>(v)dv =!: _ r<<Kv)dv m 



Jo .• ' J Jo v ' 



f v$(v)dv = k', f = k'-Cvcj>(v)dv. 

 Jo Jo 



ty(x), ^r'(x) will thus contain, besides the constants k, k', 

 only finite integrals. The value of \, on replacing in it for 

 the integration u and v by u'x, v'x, will be expanded into 

 a series, the coefficients containing certain definite integrals 

 easily reducible to one another. It is needless to give the 

 development of this calculation, which is very complicated; it 

 is only necessary to remark that the coefficients a 1} a 2 , a 3 , 

 &c. are all expressed as functions of k and k'. Consequently, 

 if a function is expansible according to the powers of .r, 

 contains two indeterminate parameters, and, when taken for 

 <£(.?•), satisfies the equation yfr(x) = ifr'(x), that will be the 

 solution, and completely so, especially if it is expansible into 



