560 Proceedings of Royal Society of Edinburgh. [sess. 
the expansion 
( x - A)(x - B)(cc - C) - (x - A)(x cos \ - a) 2 
-(x- B)(aj cos fji - b ) 2 
-{x- C)(x. cos v - c ) 2 + 2(x cos \ - a)(x cos p - b)(x cos v - c) 
and in the second paper for the axisymmetric determinant 
a — x V b" b'" 
b a + x c c 
b" c" a" + x c 
b c c a + x 
the expansion 
(a -x)(a' + x)(a''+x)(a'" + x) — ( a-x){a ' +x)c 2 - ( a " +x)(a'" + x)b' 2 
- {a-x)(a" +x)c" 2 - {a'" + x)(a' +x)b" 2 
- ( a-x)(a"' + x)c '" 2 - {a' +x){a" +x)b'" 2 
+ 2c'c"c'"(a - x) + 2 c'b"b'"{a' + x) + 2 c"b"'b’{a" + x) + 2 c'"b’b'\a'" + x) 
+ &'V 2 + bV 2 + b'" 2 c'" 2 - 2 b'V'c'c" - 2 b"b'"c"c'" - 2b'"b , c'"c\ 
— that is to say, the expansion arranged according to products of 
elements of the principal diagonal. 
A clause of the paper refers to the writings of Laplace, Vander- 
monde, Gauss, and Binet. 
Cauchy (1829). 
[Sur l’equation a l’aide de laquelle on determine les inegalites 
seculaires des mouvements des planetes. Exercices de Math., 
iv. pp. 140-160; or CEuvres, 2 e ser. ix. pp. 172-195.] 
The equation which Cauchy refers to in his title is exactly the 
equation with which we have just seen Jacobi occupied. Cauchy, 
however, comes upon it from a different direction, and it is no 
longer with him a cubic or quartic, hut an % tMc . 
The problem he sets out to solve is the finding of the maxima 
and minima of what we should nowadays call an n- ary 
quadric, viz., 
E xx x 2 + A yy y 2 + A zz z 2 + • • • + 2 A xy xy + 2A xz xz + • • • • 
subject to the condition that the sum of the squares of the n 
variables x, y, z, . . . equals 1. In a few lines it is ascertained 
that the equation in s, S = 0 say, whose roots are the extreme 
