1892 - 93 .] on Elimination of Sines and Cosines. 151 
Keferring to the complete eliminant of the 4th degree, given 
above, if we suppose the four coefficients whose suffixes are 8 and 9 
to become zero, every term in the Head Line is divisible by the 
original equations fall one degree, and therefore if the operation of 
forming the determinant has been correctly performed, the tenth 
and eleventh columns ought to vanish, and the determinant ought 
to fall to the 9 th order. By merely looking down the columns, as 
given, we see that this is so, and that by withdrawing the lines 5 
and 6 the diagonal from right to left furnishes a term, which 
does not vanish. 
Again, if we suppose the coefficients whose suffixes are 6, 7, 8, 
and 9 to become zero, the remaining terms in the Head Line are 
divisible by x‘^, the last four columns vanish, as they ought to do, 
and by withdrawing the lines 5, 6, 7, and 8, the determinant is 
reduced to the 7th order, giving a term a-^h^ in the diagonal, which 
does not vanish. 
Lastly, if we suppose the coefficients whose suffixes are 4, 5, 6, 7, 
8, 9 to vanish, the determinant reduces to the 5th order and degree, 
and the expression, as given, also satisfies this test. 
If we propose to find the eliminant of two equations in sin 0, 
cos 0, from a determinant of an order higher than 2 ? 2 + 1, we shall 
get an eliminant of the degree 4?^, as before, but with greater labour, 
and in a more inconvenient form. 
Thus, in the case considered of two equations of the 4th degree, 
if we begin by raising the given equations to the 6th degree, and 
then express ?/^, and in terms of the determinant will 
be of the 13th order [2(?z + 2) + l]. But we shall only be able to 
use five of the six multiples of each equation, since, if ((/>) denote 
one of the given equations, evidently the product (</>)^^, after reduc- 
tion, is identical with {<i>)z‘^ - (<fi)x^. The determinant will then 
consist of ten lines of the 1st dimension and three lines of the 2nd 
dimension in the coefficients, the eliminant being, as before, of the 
16th degree. So, if we raise the given equations three degrees, the 
determinant, after reduction, will be of the 15th order; and as we 
can only use seven of the multiples of each equation, there will be 
fourteen lines of the 1st dimension and one line of the 2nd dimen- 
sion, the eliminant being of the degree 16, or 4?z, as it should be. 
For equations of different degrees, say the 2nd and the 3rd, the 
