146 Froceedings of JRoyal Society of EdinhurgJi, [sess. 
also reduce x, and again z, to the 1st degree by such substitution, 
and obtain from the transformed primitives two other sets of 
derived equations of the 2nd dimension and degree ; the number 
of possible derived equations being accordingly Zn - 3. But these 
derived equations are not all independent, and the right way of per- 
forming the elimination is to use only the n-1 derived equations of 
one of the sets. Supposing we make use of the derived expressions 
of the set LM' -L'M = 0, &c., formed as above directed from equa- 
tions from which all powers of y except the 1st have been removed : 
after removing from the derived expressions, these are to be 
multiplied by y and or y and z ; and we have then 2w - 2 derived 
equations, being of the degree wH-1, and of the 2nd dimension in 
the coefficients. 
But as the equations only contain the 1st power of y, the order 
of the required determinant is, as already seen, 2?z-}-l. Such a 
determinant may be formed from the six multiples of the original 
equations, and 2?i - 3 of the 2?z - 2 derivatives. Also, as the derived 
equations are of the 2nd dimension, the degree of the eliminant is 
6 + 2(2?^ - 3)[ = 4?i], for any two equations of the like degree, n, in 
sin 6 and cos 0. 
In order that we may be assured that the derived equations 
inserted in the determinant are independent, so that the eliminant 
shall not vanish identically, the following condition suffices : — 
If we denote by y the quantity which is reduced to the 1st power 
by preliminary transformation, then, each of the derived 
equations is to be multiplied by y, and in these multiples y^ is to be 
replaced by ^2 . secondly, the same derived equations are to be 
multiplied by x or z, symmetrically. 
The manner of forming the derived equations, and the selection 
of the most suitable multiples for insertion in a symmetrical manner 
in the proposed determinant, will be more readily apprehended from 
the following example, in which I shall find the eliminant (16th 
degree) of two complete equations in sin 6 and cos 9 of the 4th degree. 
I shall suppose the 2nd, 3rd, and 4th powers of y to be removed 
by expressing y^ and y^ in terms of z^ - x^, and the equations to be 
given in the form shown in the first page of this paper. In order 
to form the required derivatives, the original equations may be 
written in the three forms — 
