146 
PROF. H. F. BAKER ON CERTAIN LINEAR 
in which X, Y respectively denote g—x and q—y, and also 
(» 2 -f 3 ) * = ^ [173 
cc + X.y/3 + Y 
y+2 ex + 0x-[e‘- j£)y = [X-Yv/3], 
The first equations have integrals expressible by 
7 h 2 
— = 1 + 2A cos 6, ~y 
A l 
= /+ 
the point (A, 0) moving in an ellipse of eccentricity 2A and semilatusrectum l. With 
these values the other equations are much simplified if we take 9, instead of the 
time t, as independent variable, as was pointed out to me by Mr. H. M. Garner, of 
St. John’s College, Cambridge. With this change they become 
where 
and 
a — 
(l + 2A cos 0) (X"-2Y'-X)-4A sin 0 (X'-Y) = aX+KY, 
(1 + 2A cos 9) (Y // + 2X'-Y)-4A sin 0 (Y' + X) = hX+bY, 
8S-E-M 7 _ 3 (E—M) y/3 
(I.) 
4 fi 
X! 
h = 
dX 
do ’ 
Afj. 
b = 
_ — 4S + 5 (E + M) 
4/x 
X" = 
d 2 X 
de 2 
, &c. 2 
together with 
(l + 2A cos 9) (x"—2y'—x) — 4A sin 9 (x' — y) — 2x = (Xy 7 3 +Y), 
4/x 
(l +2A cos 9) (y" + 2x' — y) — 4A sin 9 (y' + x) +y = (X—Yy/ 3). 
4 ft 
(II.) 
J 
The first thing 
then is to solve the equations (I.), after which the right side in (II.) 
will be known. Considerable simplification can be introduced by change of notation ; 
/ 27T?.\ 2 / 47rtA 
Th w = exp VT 
let w = exp 
A = jf(a+b + 2) = f, H = b+2ih), K = |-(a— b—2ih), 
so that 
xj _ ■; S + mE + wfiM -for _ 3 S + w 2 E + wM yj rn 2 \ 
~~ S + E + M ’ " S + E + M ’ M t \ 9/’ 
2 _ 0 „ SE+SM+EM 
m (S + E + M) 2 ’ 
where 
