which occur in the Calculus of Invariants. 471 



The following observation from Professor Cayley will be found 

 interesting : — 



" In the case of two variables, if 



P x = [ax + by) — + {ex + dy) ^, 



then in the notation of matrices, 



whence also 



P*P,=P 2 *P 1 =i{^} 3 (, ) ,)(^|)=3P 3! 



which accords with your theorem, 



E!*E 2 * =E 2 ^E 1 * =EjE 2 * + 3E 8 *." 



ceedingly simple form 



i(i+l)(3i 2 -i-Q 

 2 8 . 3 2 . 5 9 

 and for S«, 6 the form 



63a 5 - 315^+224^ 2 + 14(K- 96 

 2 10 . 3 3 . 5 . 7 



I think there can be little doubt that the outstanding factor in Si,y 

 becomes more liable to decomposition into algebraical factors in proportion 

 as the number j-\-l becomes more separable into numerical factors, i. e. 

 in proportion as j-\-l contains a smaller number of distinct prime factors. 

 For this reason I purpose calculating Si, 7, S», 8 against the appearance of 

 the next Number of the Magazine. The nature of the roots, as regards 

 being real or imaginary in the equation S ,j=0, is also probably "well de- 

 serving of study. It is worthy of notice that in each of the irreducible 

 factors of Si, j for the values of/ above considered, the coefficients are com- 

 posed exclusively of the prime factors which enter intoj+l. It is hardly 



necessary to observe that the quantities ( n \-\\ (nA-^ ' (n-\- ') > wn en 

 expressed in a rational integral form, are the coefficients of the powers of 

 x in the series for [log (1+ar)]' 1 , when n is regarded no longer as a positive 

 integer, but as an arbitrary variable. 



