6 THE UNIVERSITY SCIENCE BULLETIN. 



Also, 



\M(1+D') \ = \M(1+C,) (i+C2)-'; ; 

 .-. \MD'\ = \M((1 + C,) (I+C2)-' -1) f . . ' 

 Hence finally by (9) 



{M{l+D) j = { Mil-hD')^ \ = -lM^vj^L>'>} = 



1 



{ M{(l+C,)(l+C2)-')' I- = 

 1 1 



as was to be proved. 



6. From (5) and (8) follows that if aD = Ci ± Cs, then 



1 *i 



\f(D,A,B, . . .C)\ = \f( (1+C\)-^ {1+02)' -1,A,B,...C)\ 

 f being as usual a power series. 



/' 

 Let XD = .2^Xi Ci. We may set /.,C = ± C,', .^./-xC, = Dj , 



according as X^ is positive or negative. Then XZ) = Di ± C/ = 

 Di + XiCi. 



••. S/(AA, ...B)!- = j/((i+Dx)>(i+Cx') > 

 -i,A, ... .5)( , 



andasj cj.(Ci',A, . . . B) f = j *!>( (1+Ci) ^ -1,A,. . .B)\ 



it follows 



>i 1 



\fiD,A, ...B)\ = \S{(1+C,y {1+D,)\ -1,A, . . .B)\. 

 and this by a repeated application leads to the following very 



general formula 



/, 



\f{D,A,...B)\ = \j{{X\{l+C-y -UA,...B) (. '^^^ 



