SOO Mr. J. J. Sylvester on an Impromptu 



subject took place, and the author has not cared to put it on 

 record, I feel myself under an obligation so to do, the more so 

 as it entirely supersedes the comparatively inelegant demonstra- 

 tion of my own which I had previously intended to publish. 



The method rests essentially on the following well-known 

 theorem given by Euler relative to the partition of numbers ; to 

 wit, that the number of ways of breaking up a number n into 

 parts is the same, whether we impose the condition that the 

 number of parts in any partition men t shall not exceed (m), or 

 that the magnitude of any one of the parts shall not exceed (m) . 

 Of this rule more hereafter — for the present to its application to 

 the matter in hand. 



Since a, b, c . , . are the roots of x^+p^af"'^ + . • . , we have 



Pi = a + b-{-c+ . . . 

 p^=: ab -\- ac + be + . . . 

 PQ=abc + abd+acd-\- , . . 



Let a + /3 + 7+ . . . =w, none of the quantities a, y9, 7 . . . 

 being greater than m, but a, /3, 7 . . . being otherwise arbitrary 

 and capable of becoming equal to any extent inter se. Also 

 let \ + fi + v+ ... =n, the number of quantities \, //., v, &c. 

 being never greater than (m), but the quantities themselves being 

 otherwise arbitrary, and being capable of becoming equal to any 

 extent inter se. By Euler's rule the number of systems 

 a, fi, y . . . 19 the same as of the systems \, /a, v, . . . , say P for 

 each. For any system X, /i., v . . . , we shall have px -Pfi -Pv — t 

 by virtue of the equations above written, expressible as th,e sum 

 of terms of the form Sa* . 6^ . c^ . . . ; it may easily be made 

 ostensible, that all the combinations of a, /3, 7 . . . subject to the 

 above prescribed conditions must come into evidence by giving 

 X,fi,v.: all the variations of which they admit ; but this is also 

 immediately obvious indirectly from the consideration, that were it 

 otherwise, linearrelations would subsist between the different values 

 ofpx 'Pu 'Pv' ' ' f which is obviously absurd. Hence, then, we 

 shall be able to express the P quantities of the form px . jo^ . . . 

 by means of linear functions of the P quantities Sa* .bf^ .c^ ...; 

 and conversely, by solving the linear equations thus arising, the 

 P quantities 2a* . ft^ . c^ . . . may be expressed in terms of the 

 quantities Px-P^-" > consequently Sfl"* .b^ .c^ . . . , where (m) 

 is greater or not less than any of the quantities y8, 7 . . . , will be 

 expressible by means of combinations Px-Pfi" -y where the num- 

 ber of coefficients Px-pu* > - i^^V number of which may become 



