114 INTEGRATION OF DIFFERENTIAL EQUATIONS. 



Hence 



is the solution required. 



It admits also of another form, which it may be worth while 

 to remark. 



m 



' 



/n -m+l \ 

 \ m V 



fj q 



- ( k~ m k 



-g + l) 



/ | "I \ /.y- .yyj |_ "I 



f^ +1 ) (-^r + 



~w+l -g+i 



Here we must make <f>(k) = k m and f(k) k m , and then 



nw-fl 



~ and a :. = #(.., 



therefore a= 



The value of ^, deduced from the development of (12), can 

 of course differ only in a factor of some function of k from that 

 which is given by (11), and it will easily appear, on comparing 



the values of a n in the two cases, that this factor is k~** +m . 



Let us now consider the case in which p is not divisible by 

 m, while p 1 is so. And let p 1 = qm. The two factors of 

 (9) on which our reduction operates, viz. 



(n m + 2 -p) (n- m + l +p), 

 may be written thus, 



{n - m + 2 + (p - 1)} {n - m + 1 - (p - 1)}, 

 or (n m + 2 + qm) (n m + 1 qm). 



