On Mutual- and Self-Inductance Series. 211 



This equation is of the type 



4(» + . y -l)y=«(»+«)(»+0 9 r, ... (4) 



the equation of the hypergeometric series, so that, employing 

 the usual transformations for this type, we obtain as equiva- 

 lent forms of (3) 



»i*i=«i(*i+«)Vi (3«) 



V^=tf 2 (*i-i)V* (3 6) 



!>,(»i + %. = *.(a»+f)Vi (3 c) 



Vfc=*(»4+i)(»«-i)y<. • • (3 a) 



in which 



yi=y 5 y2=«?y> */3=(i — *0tyj y±—%tyi 



x x—l 



a? l= =l— #, a' 2 = 1— a*, ^3=—^-, #4=-—, 



and 



3. In forming the primitives of equation (3), it must be 

 noted that 7 is, in ail cases, a positive integer, so that the 

 usual second particular solution of (4), viz., 



?/ = ^-*F(a + r -l, £ + 7 -l, 2- 7 , *), 



is no longer valid. However, when 7 is a negative integer, 

 the complete primitive of (4) may be obtained as follows : — 

 Denote by Gr(a, /3, 7, 8, x) the series 



q/3 a(q+l)/3(/3 + l) 



7« 717+W + D *+•••• 



and apply the operator 



/*=$(a+ 7 -i)-4a+ <*)($+#) 



to y=X^G(a + X, /3 + \, 7 + X, 1 + X, #). 



Then /*(*/) =\ 2 (^ + 7 -i)-^ 

 and ^^") = \(,3X + 27-2)^ + \ 2 (X + r -l) A .Xlog^, 



so that lt x==0 /* { (A + B ^jtj |=0, 



