6 THE UNIVERSITY SCIENCE BULLETIN. 



Also, 



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

 .-. \MD' \ = J M((i+Ci) (l+Co)-' -1) \ . 

 Hence finally by (9) 



'\M[1+D) ( = j M{l+D')' \ = \M^l (fc).^"f -= 



k=0 \ 

 \- I > 







]M(i+Cx)'(i+C2) '\ 



as was to be proved. 



6. From (5) and (8) follows that if >.D = C'l ± C2, then 



]j{D,A,B,...C)\ = \j{ il+C{)-^il-\-C.^)'^ -1,A,B,...C)\ 

 f being as usual a power series. 



Let aD = .^X, Ci. We may set XjC = ± C/, .J^XiQ = Di , 



according as Xi is positive or negative. Then XD = Di ± Ci' = 



Di + XiCi. 



± =1 



.-. \f(D,A, ...B)\ = \f({l+D,)'(l+Cr') > 



— I , A, . . . . B) (; , 



and as j <J>(Ci',A, . . . B) } = 1 *( (i+G) ^ -^A,. . . S) ( 

 it follows 



]/(AA, .. .B)S = S/((i+CO' C^^+DJ' -^A, ...B)(. 

 and this by a repeated application leads to the following very 



general formula 



>i 



\f{D,A,...B) \ = \fi{]T\(l+C-y -1,A,...B) \. ^^^^ 



