1896—97.] 
Dr Muir on Quaternary Quadrics. 
329 
number of derived equations is not without its compensating 
advantage, as it leaves that scope for the exercise of ingenuity 
forms, too, which it is often possible to obtain for the same 
eliminant is a matter of great interest, and the study of them is 
almost always certain to have an instructive result. This appears 
very clearly from the work recently devoted to the elucidation of 
one of Sylvester’s cases, — the first case of all, in fact, and there- 
fore classical. It consists in the elimination of x, y, z from the 
equations 
where there being six secondary variables, x 2 , y 2 , z 2 , yz, zx, xy, it 
was necessary to seek for three other equations containing the 
addition or subtraction of multiples. The three ingeniously arrived 
at were 
but, when they had been got, the resulting determinant was, of 
course, of the 6th order, and, on investigation, it proved to be 
twice the square of the real eliminant. 
3. This same case of Sylvester’s turns out now to be most 
instructive fronyi totally different point of view. On first thoughts, 
it would seem as if nothing could be easier than the generalisation 
of the problem from the case of three variables to four. Yet such 
is very far from being the truth, and an excellent illustration is 
thus afforded of another peculiarity of the method, viz., its apparent 
eapriciousness. 
Consider the four equations 
which is half the charm of mathematical work. The variety of 
same variables and not derivable from the original three by mere 
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