240 The Ohio Journal of Science [Vol. XVI, No. 6, 



The A-series is 



^" (1+ -p- x+ p^^"+ ^9233 ^^+ etc.) 



And the B -series is 



-^x ( — Y + — 10 02 ^ ' fT 12 92 02 ^ "T etc. j 



Note — Equations of the sort that give two or more solutions 

 starting with a value of m different from zero, can be reduced to 

 the latter sort in two ways. First, by a change of the dependent 

 variable. If the starting value is x'", set y = x''y. The equation 

 just considered, if we set y = x-y, becomes 



dx- dx 

 and x™ = y, gives m^ x"""^— x"' = 



• \- ^ -A 

 (m-fr)- 



Form = 0, A,= A,_J~^\ 



This gives the A-series. 

 Differentiating the m-factor, 



1 \ / -2 \ 



(- 



(m-|-r)^y Vm+r 

 becomes, for m = 0, 



The B -series is derivable from the A-series by 



1 \ ^ / 



The resulting two series multiplied by x-, give the required 

 A-series and B -series. 



Or, we might change m — k to n, and deduce the series in 

 terms of n as heretofore in terms of m. Thus case (b) is always 

 reducible to case (a). 



(c) Series starting with values of m differing by a multiple 



of s when 



00 



V -= :S ArX'"+^'^ 







is the assumed solution. 



