452 



Prof. A. Cayley on Quaniics. 



[June 20.. 



the yellow being the least marked. The red, green, and blue are how- 

 ever particularly well rendered by reflected light, and the plate shows 

 the colours as seen when a dull light is thrown on the slit of the spec- 

 troscope, a simile which was suggested to me by Mr. Norman Lockyer. 



From the evidence obtained by these experiments it appears that 

 two or three molecular groupings are sufficient to give the necessary 

 colours, a subject which I only allude to, since the more general question 

 of molecular groupings is being considered by others. 



III. " A Tenth Memoir on Quantics." By A. Cayley, Sadlerian 

 Professor of Pure Mathematics in the University of Cam- 

 bridge. Received June 12, 1878. 



(Abstract.) 



The present memoir, which relates to the binary quintic (*)(#, yY, 

 has been in hand for a considerable time ; the chief subject-matter 

 was intended to be the theory of a canonical form discovered by myself, 

 and which is briefly noticed in " Salmon's Higher Algebra," 3rd Ed 

 (1876), pp. 217, 218; writing a, b, c, d, e, /, g . . . u, v, w, to denote 

 the 23 covariants of the quintic, then a, b, c, d, / are connected by the 

 relation / 2 = — a 3 d -f- a?bc — 4c 3 ; and the form contains these co- 

 variants thus connected together, and also e ; it in fact is . . . 

 (1, 0, c,f, a% - 3c 2 , o?e - 2cf)(x, y)\ 



But the whole plan of the memoir was changed by Sylvester's 

 discovery of what I term the Numerical Generating Function (N.G.F.) 

 of the covariants of the quintic, and my own subsequent establishment 

 of the Heal Generating Function (R.G.F.) of the same covariants. 

 The effect of this was to enable me to establish for any given degree 

 in the coefficients and order in the variables, or, as it is convenient to 

 express it, for any given deg-order whatever, a selected system of 

 powers and products of the covariants, say a system of " segregates ;" 

 these are asyzygetic, that is, not connected together by any linear 

 equation with numerical coefficients ; and they are also such that every 

 other combination of covariants of the same deg-order, say, every 

 " congregate " of the same deg-order, can be expressed (and that, 

 obviously, in one way only) as a linear function, with numerical co- 

 efficients, of the segregates of that deg-order. The number of con- 

 gregates of a given deg-order is precisely equal to the number of the 

 independent syzygies of the same deg-order, so that these syzygies give 

 in effect the congregates in terms of the segregates : and the proper 

 form in which to exhibit the syzygies is then to make each of them 

 give a single congregate in terms of the segregates, viz., the left hand 

 side can always be taken to be a monomial congregate a a lP . . ., or, to 

 avoid fractions, a numerical multiple of such form, and the right hand 



