411 
1888-89.] Dr T. Muir on the Theory of Determinants. 
shall have a solid square m + n terms deep and m + n terms 
broad.” (liv. 1) 
The rest of the rule deals of course with the formation of the 
terms from this square of elements, the old and familiar method 
being followed of taking all possible permutations and separating 
the permutations into positive and negative. As applied by 
Sylvester in the case of the elimination of x between the equations 
ax 2 + bx + c = 0 
lx 2 + mx + n = 0 
that is to say, as applied to the development of the determinant of 
the system 
a b c 0 
0 a b c 
1 m n 0 
0 l m n , 
the method is lengthy. 
No hint at an explanation of this or either of the two other rules 
is given. The principle at the basis of them all, however, is 
essentially that of the preceding paper. A single example will make 
this plain, and will at the same time serve to give a better idea of 
the two remaining rules than could be got by mere quotation.* 
Let the two given equations be 
ax 3 + bx 2 +cx + d = 0 
ax i + J3x B + yx 2 + Sx + e =0 
and suppose that it is desired to obtain their “ prime derivative ” of 
the 2nd (r th ) degree, that is to say, the derivative of the form 
Ax 2 +' B# + C = 0 . 
Taking the first equation followed by m - r - 1 equations derived 
from it by repeated multiplication by x } and then the second equa- 
tion followed by n - r - 1 equations derived from it in like manner, 
we have m + n- 2r equations, 
ax 3 + bx 2 + cx + d — 0 
ax^ + bx B + cx 2 + dx =0 
ax 4 + f3x B + yx 2 + Bx + e =0 
* The third rule is incorrectly stated. 
