PROF. H. F. BAKER ON CERTAIN LINEAR 
1652 
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§ 14. Leaving aside this point, we pass on now to the application of the general 
method here explained to the computation of the integrals of particular differential 
equations with periodic coefficients, as, for instance, the equation for the motion of 
the lunar perigee, considered by G. W. Hill. 
It is known from the general theory that the solutions of the n equations 
dxjdt = 11 ^+ ...+u ln x n , (i = 1, 2, ..., n), 
in which u a , ..., u in are single-valued functions with a common period, say tv, can be 
written, in the most general case, in the forms 
x i = th 1 e Al ^ il +... + A a e A4 ^> in , 
wherein A 1; ..., A n are arbitrary constants, \ 1} ..., X n are n definite constants, and the 
functions are n 2 definite functions all with the period tv. In many applications it 
is the constants X l5 ..., \ n which it is of most importance to find; when these are all 
pure imaginaries, the motion* represented by the differential equations presents, 
beyond the fundamental period tv, secondary oscillations of periods r/i\ r , and the 
motion is conventionally said to be stable. 
We show first how this form of solution naturally arises from the point of view we 
have adopted. 
Write Qq (u) in place of Q (u), and for simplicity write only two rows and columns 
of the matrix, though the argument is quite general. Make the limitation, which, 
as is well known, does not cover all cases, that there exists a matrix of constants, h, 
of n rows and columns, whose inverse is denoted by A -1 , such that the complete 
matrix Off (u) can be written in the form 
o 0 w H = 
0 \h- 
witli only diagonal elements, here denoted by e w ' w , e lC2W , in the reduced matrix. This 
will be so, in the technical phraseology, if the matrix Q 0 W (u) has linear invariant 
factors.! Then, from the definition of L (u), 
while, as u has period w, 
Q 0 w+t (u) = Q w w+t (u) .Q 0 w {u), 
& w w+t (u) = QJ (u). 
* Interesting physical examples are given by Lord Rayleigh, 1 Collected Works,’ III., p. 1. 
t A proof of the general theorem for the reduction of a matrix, valid when this is of vanishing 
determinant, is given, ‘ Proc. Camb. Phil. Soc.,’ XII. (1903), p. 65. The literature of this matter, which 
begins with Sylvester, ‘Coll. Papers,’ I., pp. 119, 139, 219, and Weierstrass, ‘Ges. Werke,’ I., p. 233, 
is very wide. The reader may consult Mutii, ‘ Elementartheiler,’ Leipzig, 1899. 
