On the Secular Inequalities in the Planetary Theory. 267 
mitted; and it was perhaps this peculiarity that kept them so 
long unrecognized ; but careful study has led to their accept- 
ance by the foremost meteorologists. He could not hope to 
solve the complex problem as it really exists, and was obliged 
to make his solution very general. 
On pp. 386-391, Band xiv. of the est err. Zeitsch. fur Me- 
teorologie is given a review of the first part of Ferrel's elabo- 
ration, mentioned below, of his early paper; pp. 161—175 and 
276-283 of the xvii. Band of the same Journal contain a 
review of the second part ; and the whole was again reviewed 
in ' Xature ' last year. 
It appears as though Ferrel's critic had written his article 
from the point of view of a student of Laplace and Airy, and 
had not examined the more modern text-books on mechanics 
to see if Ferrel's reasoning was admissible. 
It is hoped that instead of heeding the warning that has 
been sounded against Ferrel, the readers of this Journal will 
read the elaborations of his first paper, given as Appendices 
to the U. S. Coast-Survey Eeports for 1875 and 1878. 
Mr. Heath's paper will have one good effect, I hope; and 
that is, to interest some of the English mathematicians and 
physicists in this subject. 
XXXIX. On the Equation to the Secular Inequalities in the 
Planetary Theory. By J. J. SvLYESTEii, P.B.S.* 
A VERY long time ago I gave, in this Magazine, a proof 
of the reality of the roots in the above equation, in 
which I employed a certain property of the square of a sym- 
metrical matrix which was left without demonstration. I will 
now state a more general theorem concerning the product 
of any two matrices of which that theorem is a particular case. 
In what follows it is of course to be understood that the 
product of two matrices means the matrix corresponding to the 
combination of two substitutions which those matrices represent. 
It will be convenient to introduce here a notion (which 
plays a conspicuous part in my new theory of multiple algebra), 
viz. that of the latent roots of a matrix — latent in a somewhat 
similar sense as vapour may be said to be latent in water 
or smoke in a tobacco-leaf. If from each term in the diagonal 
of a given matrix, \ be subtracted, the determinant to the 
matrix so modified will be a rational integer function of \ ; 
the roots of that function are the latent roots of the matrix ; 
and there results the important theorem that the latent roots of 
* Communicated by the Author. 
