DIFFERENTIAL EQUATIONS OF ASTRONOMICAL INTEREST. 
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appropriate to use a theorem* for the expression of a determinant of 2n rows and 
columns as a Pfaffian, a sum 1.3.5... (2n— l) terms, of which each term is a product 
of n factors, each factor being of the form 
(12) = a 1 b\ - a\b x + a 2 b' 2 - a' 2 b 2 + ... + a n b' n - a' n b n , 
where the elements 
ci\b\, a 2 b 2 , ..., ct n b n , 
a'l&'i, a' 2 b' 2 , ... a' n b' n , 
are the constituents of two rows of the determinant. For in this case the factors (12) 
are easily calculated. But we do not pursue this method. 
§ 9. Instead we proceed as follows. In the equations 
[l+\(i + ^ 1 )][x"-2y'] = px, 
[1+X(f+^- 1 )] \_y" + 2x'~\ = qy, 
where f = e i<3 , write 
x = e U6 X, y = e ix6 Y, k = <r + * 2 X 2 + /c 4 X 4 + ..., 
in which k 2 , k 4 , ... are certain functions of p, q to be determined. Then the equations 
become 
[1 + xa+r 1 )] [X"— 2 Y' + 2bc (X'—Y) —/c 2 X] = pX,~) 
[l + X^+r 1 )] [Y"—2X' + 2\k (Y' + X)-k 2 Y] = qY, J 
which by the general theory are capable of periodic solution when k is properly 
chosen. Put then 
X = 1T \<f>i + Y(p 2 +..., Y = P (1 + + Y\fs 2 + ...), 
where P is a constant; the differential equations then take the forms 
(1 + \w) (H 0 + xHi + Xffl 2 + ...) — pX, 
(l + Xw) (K 0 + XK 4 + X^Ia^t-...) = qY, 
w denoting £+£ _1 . Comparing the coefficients of like powers of X, 
Hu =p, K 0 = P 3 , Hf+wHo = p<p u Kj + wKo = gP^n, 
and, in general, 
H n +wK n _ x =p<Pn, K n +wK n _! - P q\Js n , 
so that 
Hi = p (fa-w), K x = P q (fa-w). 
* Proved in Scott-Mathews’ ‘ Determinants ’ (1904), Chap. VIII, p. 99, § 19. Also in Baker, 
‘ Multiply-periodic Functions,’ p. 314. 
