, 04 MAJOR P. A. MACMAHON: MEMOIR ON THE 



Since L. = 0, when > /x, the series may be continued, 



and for values of / ranging from /i+ 1 to mn, 

 . GF(( , ,, .,. (25+i> GF ( ( _v m , .)+ ... 



the series having 1+ 1 terms ; but, when I > mn, 



the series having m + 2 terms. 



Art. 27. We have thus a number of difference equations satisfied by the functions 

 GF (/, m, ), and we can now show that if 



GF(/,m,n) = |LM|,|LM|, ... |LM|. = J (/, m, n) 



for all values of I not exceeding mn, the law is true universally. 

 For J (I, m, n) is of the form 



P.-P^ + PjS*- ...(-) m P mn X lm , 



where the coefficients P are functions of x independent of /. 

 Then 



fnmi which it appears that 



(1_0) (i-fe) ... (1 -fe''')ij (/, m> n ) ff 







is of degree mn in 0, at most, and hence, when I > mn, since 



. 



v 1 ) I 1 )! 2 ) 



we have 



. w,n)- H!+!)j (I- !,,,)+ ... +(-)"-+' J; 1/ (" +1 'J(i-m-l,w,n) = 0; 



but it has been shown that / > mn 



GF(/, m, n)- (1H+1)GF(J-1, m, n)+ ...... +(-)"' +1 a;'>'<'^>GJ(/-mn- 1, m, n) = 0. 



