198 Mr. J. Cockle on the Theory of Equations 
78. The elimination of b between these results may be ex- 
pressed by the determinant 
a; Ba?, ya', 
Ba’, da‘, ed, 
yas, €a, fa, 
in which a, 8,.,, € are functions of a and of 3. This deter- 
minant is of the form (3, a°)a?, and rejecting the factor a, the 
result of the elimination of will be of the form 
x(S, a°) =0. 
79. A result of the same form will be obtained if we eliminate 
6 between (g’) and (e’) or (f') ; for © can only appear in the final 
results of elimination under the form @®, otherwise we should be 
led to equations one side of which would have five times as many 
values as the other. 
80. Further: the formule of Euler and Bezout are not affected 
by the binary interchange (4 a) ( c) , and we obtain, at pleasure, 
four systems of relations, which, for brevity, I shall write 
(a, b, 3)=0, (d, Cc, 3) =0, (e, a, —n) =0, (6, d, —3)=0; 
and these systems show that a and d are inseparably connected 
in the formule, and that the ultimate results will assume the 
form of quadratic equations. And such is the form which the 
equations in u and v (art. 44) indicate. 
81. Let, then, 
a°—2a°+3°=0 
denote the result of eliminating 4, ¢, and d from the equations of 
art. 75. This result is equivalent to 
©10— 20° + 51°95 =0, 
and, solving as for a quadratic, we find 
apt V EROS, 
or, a8 we may write it, 
O=y+ V V. 
82. That X and pw are rational functions of 3°, follows from the 
consideration that 
2y= (01) + (O°), 
and consequently that «and 3 are “similar” functions. Hence 
we may express 4 in terms of 3 by the process appropriate to 
such functions, or we may adopt Lagrange’s method of division, 
