[NOTE ON A THEOREM IN MATRICES.] 
[From the Proceedings of the London Mathematical Society, vol. xxn. (1891), p. 458.] 
Prof. Cayley remarks that a “ simple instance [of the theorem] is that, if the 
real symmetric matrix 
(a, h, g ) 
h, b, f 
9> f> c 
has two latent roots each = 0, and therefore a vacuity = 2, then it has also a nullity 
= 2 [which may be shown as follows], viz. the conditions for a vacuity = 2 are 
a, h, g = 0, bc + ca + ab — f 2 — g 2 — h 2 = 0, 
h, b, f 
9> f c 
or, if as usual the determinant is called K, and if 
(A, B, C, F, G, H) = (be —f 2 , ac-g 2 , ...), 
K = 0, A+B+G= 0, 
these equations give 
BG — F 2 = Ka = 0, AC-G 2 = Kb = 0, AB - H 2 = Kc = 0, 
then, if 
i.e. 
and therefore 
BG = F 2 , AC= G 2 , AB = H 2 - 
A {A + B + G) = A 2 + H 2 + G 2 , 
B(A+B + C) = H 2 + B 2 + F 2 , 
C(A + B + C)=G 2 + F 2 + C 2 -, 
or, if A + B + G = 0, then for real values 
A=B = G = F=G = H= 0, 
i.e. nullity =2.”