706 ME. A. CAYLEY ON THE EESULTANT OE A SYSTEM OE TM'O EQUATIONS. 
Next to explain the second method, viz. the calculation of the resultant from the 
expression in the form of a determinant. 
Taking the same example, as before, the resultant is 
a, b, c, d \ 
a, b, c, d, I 
2 ), q, r I 
12 q, r 
2>, q, r, ! 
which may be developed in the form 
+12.345} 
-13.245} 
+14.235'! 
+23.145J 
-15.234'! 
-24.135J 
+25.134'! 
+ 34.125J 
-35.124} 
+ 45.123} 
where 12, 13, &c. are the terms of 
( a, b, c, d ^ 
I a, b, c, d I 
and 123, &c. are the terms of 
( JP. q^ ) 
i?, q, r 
2 ), q, r 
viz. 12 is the determinant formed with the first and second columns of the upper matrix, 
123 is the determinant formed with the first, second and third columns of the lower 
matrix, and in like manner for the analogous symbols. These determinants must be 
first calculated, and the remainder of the calculation may then be arranged as follows : — 
