which occur in the Calculus of Invariants. 469 



inductively by aid of the identical equation 



*(i 0S{ i-*)y=-<^-*)T\ 



ax j. — x 



combined witb the relation 



^n+j-l,j : =^>i+j-2 ) j+J $>i+j-2j-lf 



= $n+j-2,j +j S>i+j-3,j- 1 +j(j— 1) $n+j-4, ;-2 



+JU-W~2) S rt+i - 5j -3+&c. 



It is obvious that the coefficients of the powers of x in the 

 above expansion must be all of them integral functions of n, and 

 must also contain n in every term except the first ; and when so 

 expressed as integer functions of n, the result obtained on the 

 supposition of n being a positive integer will continue to subsist 

 for all values of n. From the first part of this statement, it fol- 

 lows that $ij may always be expressed under the form 



((t+i)».(i-i)..,(i-y+i)> y _ 1 (i) 1 



where </>y_i (i) is a quantic in i of the degree ^' — 1. 

 Furthermore, if we suppose 



*. .(i)= 'fo-'W , 



Yj-i v J 2 a . 30. 5 Y . 7 s . IV. . .pQW. . . 



p being any prime number, and i/r a function of i of the degree 

 (j—1) all whose coefficients are integer, and (consistently with this 

 being the case) as small as they can be made, there is no diffi- 

 culty in obtaining the value of <p(p) under the following form, 



where, as usual, the symbol E signifies that only the integer part 

 is to be preserved of the number upon which it acts. The value 

 of the coefficient of i J '~ l in <f>j-\i is easily ascertained to be 



r, and consequently the coefficient of fr'* 1 in -v/r is 



f The equation in differences $ n j = S n -i,j-{-jSn--i,j-i gives an easy- 

 algorithm for calculating S«, j, and shows a priori that it is divisible by 



(n+l>...(rc-j+l). 



pj 

 X Consequently 4>(p), the exponent of p, is always less than , __-. x^and 



ip-iy 



a fortiori than -J- — , 

 p-2 



