134 
PROF. H. F. BAKER ON CERTAIN LINEAR 
the absolute term in the coefficient of X 4 in this being 
2 K i ((t 0 O" & 0 ) “t + 2iQ,20 K 2 cr “t + 2b m K 2 + 2& 40 (X + C 4 Q. 
Thus, as in § 1, we first put 
a 0 (r 2 +2b 0 or + c 0 = 0, 
assuming, as in § 1 it was assumed that vju is not an integer, that 
2 (a 0 <r + b 0 )/a 0 , or 2 (« 0 c 0 - bfffa 0 , 
is not zero or integral; then the absolute term in the coefficient of X 2 determines 
*2 («o°' + ^u)> and hence k 2 , and the absolute term in the coefficient of X 4 similarly 
determines k 4 . 
The excepted case in which k contains odd as well as even powers of X we may 
leave aside at present. 
§3. We may apply the preceding to the much discussed* equation 
d 2 x 
When X = 0 we have the two factors e iai , and the general case is that in 
which e 2,M has not the period, 2tt, of the coefficients in the differential equation, 
that is, when 2<r is not an integer. First assume this to be so. Then writing 
we obtain 
x = ff* e X 
X" + 24X' + (<r 2 —/c 2 + 2\lc 1 cos B + 2\% cos 20+ ...) X = 0. 
Herein assume 
K = <7 + AToX"' + /C 4 X 4 + . . . , X — 1 + \<p\ + X“02 + • • • ) 
* For this differential equation the following list of references may be useful, though it is probably far 
from complete: — Mathieu, ‘Louville’s J.,’ XIII. (1868), p. 137; Hill, ‘Coll. Math. Works,’I., p. 255 
(‘Acta Math.,’ VIII. (1886)); Adams, ‘Coll. Scientific Papers,’ I., p. 186, II., pjx 65, 86; Tisserand, 
‘ Mec. CM.,’ t. IIP, Ch. I. ; Poincare, ‘ Meth. Nouv.,’ t. II., Ch. XVII. ; Forsyth, ‘Linear Differential 
Equations’ (1902), p. 431; Rayleigh, ‘Scientific Papers,’ vol. III. (1902), p- 1; Lindehann, ‘Math. 
Annal.,’ Bd. XXII. (1883), p. 117; LlNDSTEDT, ‘ Astr. Nachr.,’ 2503 (1883); Lindstedt, ‘ Memoires 
de l’Acad. de St. Petersbourg,’ t. XXI., No. 4; Bruns, ‘Astr. Nachr.,’ ^533, 2553 (1883); 
Callandreau, ‘Astr. Nachr.,’ 2547 (1883); Callandreau, ‘Ann. Observ.,’ Paris, XXII. (1896); 
Tisserand, ‘Bull. Astr.,’ t. IX. (1892); Stieltjes, ‘Astr. Nachr.,’ 2602, 2609 (1884); ILarzer, 
‘Astr. Nachr.,’ 2850, 2851 (1888); Moulton and Macmillan, ‘ Am er. J.,’ XXXIII. (1911); 
Moulton, ‘Rendic. Palermo,’ XXXII. (1911); Moulton, ‘Math. Ann.,’LXXIII. (1913); Whittaker, 
‘.Cambridge Congress’ (1912), L, p. 366 ; Whittaker, Young and Milne, ‘Edinburgh Math. Soc.,’ 
XXXII., 1913-14; Ince, ‘Monthly Not.,’ Roy. Astr. Soc., LXXV. (1915); Poincare, ‘Bull. Astr.,’ 
XVII. (1900). 
