438 Prof. Sylvester on the Integral of 



and read them off respectively into products as below : — 



4.3x3.2x2.1x1 .0 

 (4.3x3.1xl.0)x(-2.2) 

 (4.2x2.1xl.0)x(-3.3) 

 (4.3x3.2x2.0) x (-1.1) 

 (4 . 3 x 3 . 0) x (2 . 2 x 1 . 1) 

 (4.2x2.0)x(3.3xl.l) 

 (4 . 1 x 1 . 0) x (2 . 2 x 3 . 3) 

 (4.0) x (1.1x2. 2x3. 3) 



The sum of the above terms is the value of the determinant 

 in question. And so in general, if we define u n by means of the 

 equation 



(n . n)u n + (n . n-—l)u n _ 1 -f (n . n — 2)w w _ 2 -f • • • = ; 



with the initial conditions as above stated, the value of u n to a 

 factor pres will be represented by 



£(», n Xi n 2) ... w w , 0), 



where n >n v > n 2 . . . >w u [g) = 0, 1, 2, . . . (n— 1)] and (n } n v n 2 , 

 . . . n^, 0) is to be interpreted as meaning 



M.xn.n l xn l .n <2 x ... x n^ . 0, 



where to find M we write the complementary integers 



m l> m 2> m 3) ' * • m n—u + lt 



which together with n v n Q , . . . ra w make up the complete tally of 

 all the integers from 1 to (n—Y), and then write 



M=( — )»-«+i (m^nii) . {m 2 .m 2 ) . . . (m n _„ +1 . m n _ w+1 ). 



In order to form by an exhaustive process all the descending- 

 series above described, we may if we please consider the differ- 

 ences of the terms of any such series, and write 



B = n — n v 8 l =n l — w 2 ...S w = w w , 

 we have then 



8 + 8 1 + 5 2 + . . . +8 a) =xw. 



So that the question is reducible to that of finding all the parti- 

 tions of n, and of permuting in every possible manner the terms 

 in each such system of partitions ; for it is obvious that in general 

 the value of (n, n D ?z 2 , . . . n M , 0) depends not only on the mag- 

 nitudes, but on the order of sequence of S 9 S lt 8 2 , . . . 8 W . 



If we suppose that the order of the differences is limited, as, 

 for example, that the equation is of the ith order, then any such 

 coefficient as r . s is to be considered as zero when r un s > i, and 

 consequently the partitions of n are to be limited to parts none 

 greater than i. Moreover, if in such case the coefficients become 



constant, so that r . 6 = </>(?• — s), it is apparent that the order of 



