470 Prof. Sylvester on the Properties of the Test Operators 



always an odd number, the number of times that 2 is contained 

 in this denominator being 



v^ ^ 3 



The maximum prime in the denominator of the fraction which 

 expresses $j-\ (i) enters always as a simple factor, because, as we 

 know by M. Bertrand's theorem, there is always a prime number 

 included between q + 1 and 2^ + 2. Consequently, supposing^' 

 to be 2q or 2^ + 1, since there exists a prime number p greater 

 than q + lj and not greater than 2q + l t this prime number 

 will appear in the denominator of B n ,j with the exponent 



E — - — — 4r , i. e. unity. 



Conversely, if by any means not founded on the above theorem 

 we could ascertain this fact, we should be in possession of an 

 entirely new proof of that celebrated theorem. It is perhaps 



also worthy of a passing notice, that (— ) j • ^_i) (,/ — 1) mayeasily 

 be proved to be equal to the coefficient of /^ in log log (1 + 1) — log^f. 

 I have calculated the values of s it i' f s i)2 ', s i>3 ; s if 4 , which are 

 as follows : 



(t+i)i . (i+i)i(«-i) . (t+i)i(.--i)(i-g ) . 



(i+1) % l }:~ 5 2)ii ~ S) (^i s -^+ioi-s)t. 



t And more generally if 



s M +i-i,i=(( M +i)(^+i™i) • • • »)(Ofif- l +o^ l nt-'+ . . . +C0, 



(_yC w = S. coefficient of lin (i og ( lo g ( 1 +0) V. 



% In his great and most useful work on the Calculus (p. 264), Pro- 

 fessor De Morgan has applied Arbogast's method to the expansion of 

 (log (I+x)) n , and worked out his results completely as far as the co- 



C E F 



efficients of a; 4 inclusive. His — , — , — , when i—1, i-% z—3 are substi- 

 tuted in these quotients for n, become identical with the non-trivial, or so 

 to say outstanding factors in my S»,2; Si, 3,' Si, 4 respectively. 



I have since calculated the same factors for Si, 5 , S;, 6 corresponding to 



P TT 



Professor De Morgan's — , — , when n is replaced by i-4,i-5 respectively. 



The calculations are rather laborious, extending in the latter case to 8 

 places of digits ; but comparatively very sm°ll numbers appear in the 

 final expressions. For Si, 5 I find the outstanding factor takes the ex- 



