34] NOTE ON THE MAXIMA AND MINIMA OF FUNCTIONS OF THREE VARIABLES. 229 
It is required to show that if A' + B’ + G' and A" + B" + G" are positive, A', B', C', 
A", B", G" are so likewise. 
Consider the cubic equation 
(A' - k) (B' - k) (O' — k)~ (A' - k) F' 2 - (B' - k) G' 2 - (C - k) H' 2 + 2F'G'H' = 0, 
the roots of which are all real. By the formulas just given this may be written 
k 3 -k 2 (A' + B' + C / ) + k(A" + B'' + C' , )-K 2 = 0; 
and the terms of this equation are alternately positive and negative; i. e. the roots are 
all positive. Hence the roots of the limiting equation 
(B'-k) (G' — k) — F' 2 = 0 
are positive, i. e. B' + G' and B'G' are positive: but from the second condition B', C' are 
of the same sign: consequently they are of the same sign with B' + C', or positive. 
Also A" = B'G' — F' 2 is positive. Similarly, considering the other limiting equations, 
A', B', C', A", B", G" are all of them positive. 
In connection with the above I may notice the following theorem. The roots of the 
equation 
(A - ka) (B - kb) (G - ok) - (A - ka) {F - kf) 2 - (B - kb) (G - kg) 2 - (C - kc) (H - kh) 2 
+ 2 (F- kf) (G - kg) (H - kh) = 0, 
are all of them real, if either of the functions 
Ax 2 -f By 2 + Gz 2 + 2Fyz + 2Gxz + 2Hxy, 
ax 2 + by 2 + cz 2 + 2 fyz + 2 gxz + 2 hxy, 
preserve constantly the same sign. The above form parts of a general system of proper 
ties of functions of the second order.