164 
PROF. H. F. BAKER OK CERTAIN LINEAR 
putting therein t — 0, so obtaining, say A 0 , equal in our notation to hyhr x , and then 
solving the determinantal equation 
IA 0 —A | = 0, 
whose roots are ic x and ic 2 . This process will be found to be equivalent to the 
general procedure explained by Laplace, in the passage above referred to, for 
bringing tire time under trigonometrical signs. AVe have considered only the case 
of linear differential equations with periodic coefficients, and have supposed Q 0 W (u) to 
have linear invariant factors ; Laplace’s method, if less definite, is of much wider 
application. An interesting exposition of the method in general is given by 
M. 0. Callandreau, ‘Ann. de l’Observ. de Paris,’ XXII., 1896, pp. 16, 20. 
AA'e may notice that 
A 0 = h/ic 1 , 0\/? _1 
0, ic 2/ 
gives 
so that we also have 
Q (A 0 ) = h /e ic '\ 0 \ h~\ 
\ 0, e ic V 
Q ( u ) = P(/ . ii (A 0 ) 
— Po f (1 + tA 0 + \t 2 A 0 2 +...), 
and the quantities e' ClW , e lClW are the roots of the equation 
\Q 0 w (u)- P \ = 0. 
§ 15. When the sum of the diagonal elements of the matrix u is zero, the 
determinant of Q ( u ) is unity, as above remarked. In this case, when n — 2, the two 
quantities e lc ' w , e' C2W are inverses and c 2 = — c x . In this case the equation 
\Q 0 w (u)- P \= 0 
gives at once the value of cos cw. This appears, however, a less advantageous way 
of determining c t , c 2 than that explained above, as requiring greater approximation 
in the calculation of { u )> as will be seen in examples. 
The fact that c x , c 2 are equal and of opposite signs is a particular case of a 
well-known theorem for the variational equations arising in the general dynamical 
case, which is proved by Poincare (‘Meth. Nouv.,’ I., 193). The following proof, 
though longer, appears more fundamental in character. The general dynamical 
equations being 
dx r r/F dy r 
