April, 1916] Derived Solutions of Differential Equations 237 



1 



If we differentiate the m-factor, — z^, ^2' occurring m 



(2), we have 



/I 



y (m + r)^ / \m+r 

 And for m = 0, this is 



And the n*'' term of the B-series comes at once from the 

 (n + l)*'' term of the A-series (beginning with the 2d term of 

 the A-series) by multiplying by 



n 

 S 

 1 

 So that the B-series is 



X „ X 



11 x^ 25 . x^ 



12 i-:2- 3 1-. 22.32 6 12.22.32.42 ^ 



As this method of getting the B-series from the A-series 

 gives the relation of corresponding terms, it makes the settle- 

 ment of the question of the convergency of the B-series much 

 easier than Johnson's method. Into the question of con- 

 vergency I am not entering here, but merely showing how to 

 get the B-series, whether or not it is a usable solution. 



The reason for the above procedure is this: The A's come 

 each from the preceding by multiplication: 

 Ai = Ao(l+gi(m-k)+ etc.) 

 A2=Ai(H-go(m-k)+etc.) 



= Ao(l + (gi+g2) (m-k)+etc.) 

 A,3 = A2(l+g3(m-k)-fetc.) 



= Ao(H-(gi + g2 + g3) (m-k)+etc.) 

 and so on. 



n 

 The coefficient of m — k in An is S g„. 



And since, in the expansion, the coefficients of m — k are values 

 of F'(m), the B-series comes from the A-series by the 



n 

 relation 2 F'(m) using the m factor of Ar-i, as in the fore- 

 going example. 



(2) x(l-x2)4^-f (1-3x2)^- xv=0 (Johnson-lSl) 

 dx- dx 



y = x"> gives m2x"^-i-(m+l)2 x'"+i = (E) 



