118 



MONTMOET. 



Nicolas Bernoulli then gives another form for these expressions ; 

 we will exhibit that for z from which the others can be deduced. 



Let 



^^i, ,= K ,= !i. Then 

 ^ n \sj m 



z = aq(l-r)-\-hq\l-r{l-\-tl]\-\-C(i\l-r 



r _ fia-r)~\] 



-{■dq \l — r 



-^ ^ ^^ ^ tH{i-i) ^ fi{i-i)(i-2y 



1.2 



1.2.3 



I • • • J 



this series is to be continued for I terms. 



The way in which this transformation is effected is the follow- 

 ing : suppose for example we pick out the coefficient of c in the 

 value of z, we shall find it to be 



1 .Zs [ s s s'^ ) 



where the series in brackets is to consist of Z — 2 terms. 

 We have then to shew that this expression is equal to 



«{'-'h"*'^']}- 



We will take the general theorem of which this is a particular 

 case. Let 





where 



Let 



P.= 



p + X-1 



u 



1 + - + -T + 





then S=T— -y-^ . 

 [Xdmr' 



Now 



- fmr 



u — 



1- 



la 





say; 



