DIFFERENTIAL EQUATIONS OF ASTRONOMICAL INTEREST. 
185 
Mr. E. Lindsay Inge, of Trinity College, Cambridge, following up the method of 
his paper referred to above (footnote, p. 13 4), has calculated numerical results for the 
case when Jc u Jc 2 , ... have the values considered by C4. W. Hill.] 
PART III. 
§ 22. I desire to add to the foregoing some very incomplete remarks in regard to a 
generalisation of which the work appears to be capable. The most important general 
result obtained is that when u is a periodic matrix, the matrix 12 ( u ) can be expressed 
as a periodic matrix P multiplied into a matrix involving quantities of the form e AT . 
One direction in which this result can be amplified is by extending the assumption 
we have made that the matrix Q™ (u) has linear invariant factors. It is well enough 
understood what is the character of the modifications thereby introduced. A more 
important generalisation appears to be that the factorisation of the matrix 12 (u) 
does not in fact require that u be a periodic matrix. As an indication of the theorem 
consider an equation 
/72/y, 
— +cr 2 x = x (ae iKt + be iKt + ce 11 * 1 ), 
dt 2 
in which the constants k , A, /x are such that /c + A + /x = 0, but the ratio of two of them 
at least is irrational. For example, we might have k = v2 + l, u= — ^2+1, 
/j. — — 2. Then, assuming that there exists no identity of the form 
Cite + /3A + yfjL + 2cr = 0, 
in which a, 8, y are positive integers, the equation would seem to have a solution of 
the form 
x = e , 2 f X, 
where X is a series of positive and negative integral powers of e lKt , e lM , e iat , which 
may be arranged as a power series in a , b, c, and q is a series of the form 
q = ar + K x abc + A 2 a 2 b 2 c 2 + ..., 
in which A 1; A 2 , ... are constants. The differential equation has not periodic 
coefficients. 
In a paper already far too often referred to, ‘ Proc. Lond. Math. Soc.,’ XXXT., 
1902, p. 353 et seq., replacing the variable there called t by e T or Q it is shown 
(p. 365) for the equation system 
~ = (A + t;V)x, = ux, say, 
d-r 
in which A is a matrix of constants, and V a series of positive integral powers of C, 
that there is a factorisation of the matrix 12 (u), in the form P12(<^)y, where P is a 
2 c 
VOL. COXVI.-A. 
