268 On the Secular Inequalities in the Planetary Theory. 
any function of a matrix are respectively the same functions 
of the latent roots of the matrix itself : ex. gr. the latent 
roots of the square of a matrix are the squares of its latent 
roots. 
The latent roots of the product of two matrices, it may be 
added, are the same in whichever order the factors be taken. 
If, now, m and n be any two matrices, and M = mn or nm, I 
am able to show that the sum of the products of the latent 
roots of M taken i together in every possible way is equal to 
the sum of the products obtained by multiplying every minor 
determinant of the ith. order in one of the two matrices to, n by 
its altruistic opposite in the other: the reflected image of any 
such determinant, in respect to the principal diagonal of the 
matrix to which it belongs, is its proper opposite, and the 
corresponding determinant to this in the other matrix is its 
altruistic opposite. 
The proof of this theorem will be given in my large forth- 
coming memoir on Multiple Algebra designed for the 'Ame- 
rican Journal of Mathematics.' 
Suppose, now, that m and n are transverse to one another, 
i. e. that the lines in the one are identical with the columns in 
the other, and vice versa, then any determinant in to becomes 
identical with its altruistic opposite in n; and furthermore, if 
to be a symmetrical matrix, it is its own transverse. Con- 
sequently we have the theorem (the one referred to at the 
outset of this paper) that the sum of the t'-ary products of the 
latent roots of the square of a symmetrical matrix (i. e. of the 
squares of the roots of the matrix itself) is equal to the sum of 
the squares of all the minor determinants of the order i in the 
matrix; whence it follows, from Descartes' s theorem, that when 
all the terms of a symmetrical matrix are real, none of its 
latent roots can be pure imaginaries, and, as an easy inference, 
cannot be any kind of imaginaries; or, in other words, all the 
latent roots of a symmetrical matrix are real, which is Laplace's 
theorem. 
I may take this opportunity of stating the important theorem 
that if \ 1} X 2 , . . . Xi are the latent roots of any matrix to, then 
v (»i—\ 2 )(m—\ 3 ) . . (m —Xj) 
* w = 2 (x 1 -x, X x 1 -x 3 )...(x 1 -x/ x - 
This theorem of course presupposes the rule first stated by 
Prof. Cayley (Phil. Trans. 1857) for the addition of matrices. 
When any of the latent roots are equal, the formula must 
be replaced by another obtained from it by the usual nie- 
1 
thod of infinitesimal variation. If f^ni — m'°, it gives the 
