'88 MONTMORT. 



, (;>-g)(p-<7-i)-'-i I 



If ^ = 2 the last term within the brackets should be doubled. 

 Again if q is odd we can by successive use of the fundamental 

 formula make u^ depend on w^^^, and if q is greater than unity it 



can be shewn that u.^, = ^-^^ -77 . Thus we shall have, if a is an 



^^^ q+1 Z 



odd number greater than unity, 



,, _ g(^-^) 1 J f 1 4. (p-^)(p-g-l) 

 ^^-^(^-1)2^|'+ (^-2)(p-3) 



(p-g)(j9-g-l)(j9-g-2)(;.-g-3) 



(^_2)(^-3)(;.-4)(2.-5) 



(i>-g)(;>-g-i)...2 



"^ (i5-2)(i>-3) ^ 



If ^ = 1 we have by a special investigation Up = — . 



If we suppose q even and p — q not less than q — 1, or q odd 

 and p —q not less than q, some of the terms within the brackets 

 may be simplified. Montmort makes these suppositions, and con- 

 sequently he finds that the series within the brackets may be 

 expressed as a fraction, of which the common denominator is 



{p-2)(p-S)...{p-q + l); 



the numerator consists of a series, the first term of which is the 

 same as the denominator, and the last term is 



fe-2)(^-3)...2.1, or (^-l)(^-2)...3.2, 



according as q is even or odd. 



The matter contained in the present article was not given 

 by Montmort in his first edition ; it is due to John Bernoulli : 

 see Montmort's, page 287. 



