184 Prof. J. J. Sylvester's Proof of the undemonstrated 



O^D^ may be termed a differentiant p removed (as we speak 

 of cousins once, twice, &c. removed) from x, which will be the 

 same thing as O q T) y (a differentiant q removed from y) if p + g 

 is equal to the degree, viz. ij — 2w. Now all these differen- 

 tiants, whether simple or mixed, possess a wonderful property, 

 which may be deduced by means of Salmon's Theorem, given 

 in the Philosophical Magazine for August 1877. They are 

 all, in an enlarged sense of the term, Invariants — in this sense 

 to wit, that if the elements are made to undergo a substitution 

 consequent upon or, as we may say, induced by a general 

 linear substitution impressed on the variables, which for greater 

 simplicity of enunciation may be supposed to have unity for 

 the determinant of its matrix, then every differentiant, whether 

 single or double (the latter being equivalent to an invariant), 

 and whether simple or mixed, will remain a Constant Func- 

 tion of the Coefficients of the impressed substitution. To wit, 

 if the differentiant be p removes from x and q removes from y 

 (so that its degree is p + q\ and if the impressed substitution 

 be lx + \y for x, and mx -f /my for y, where lfJL—\m=l, then 

 will the differentiant be a constant bipartite quantic in the two 

 sets of coefficients I, m and \, //,, of the degree q in the former 

 and p in the latter — a theorem which amounts almost to a 

 revolution in the whole sphere of thought about Invariants. I 

 have borrowed the term " Imaginative Reason " from a recent 

 paper of Mr. Pater on Giorgione, in which, as in many of 

 those of Mr. Symonds (I will instance one on Milton in par- 

 ticular), I find a continued echo of my own ideas, and in the 

 latter many of the very formulae contained in my ' Laws of 

 Verse,' where versification in sport has been made aesthetic 

 in earnest. Surely the claim of Mathematics (its "Anders- 

 streben ") to take a place among the liberal arts must be now 

 admitted as fully made good. Whether we look to the ad- 

 vances made in modern geometry, in modern integral calculus, 

 or in modern algebra, in each of these a free handling of the 

 material employed is now possible, and an almost unlimited 

 scope left to the regulated play of the fancy. It seems to me 

 that the whole of aesthetic (so far as at present revealed) may 

 be regarded as a scheme having four centres, which may be 

 treated as the four apices of a tetrahedron, viz. Epic, Music, 

 Plastic, and Mathematic. There will be found to be a common 

 plane to every three of these, outside of which lies the fourth ; 

 and through every two may be drawn a common axis opposite 

 to the axis passing through the remaining two. 



So far is certain and demonstrable. I think it also possible 

 that there is a centre of gravity to each set of three, and that 

 the lines joining each such centre with the outside apex will 



