50 A NEW METHOD OF DETERMINING 
It may be of value to have y’ in terms of z" and the J functions. 
ond of equations (20) we have 
(0) (a) (2) 
yPs=—AaAt+t+ J, .2 +43,.2 +43,.2 + ete. 
(2) oO . (Or 
== dh oF linn ain Oe: 
(0) @ (2) 
yt=—Aat J .et+4dy.2°% 4+ 4d, 2% + ete. 
(2) (3) (4) 
2 
— J,.zg —td,.2? —t4d,.2? —ete. 
- (1) (0) | iO . 
Op —2SJ,.2 +2d,.2 + 2J,.2 + ete. 
@ Oates eae 
—2J, .2*— 2S, . 2° — 2d, .27* — ete. 
ze Pace bee ee) Maa. ch He 
ye = — id, 27 + 3dn. 27 + 2S3,.27 + ete. 
(3) on oO 
— fd, .2 — gdn.27 —2J3,.2° — ete. 
Then from 
y' + y~ = 2 cos ve 
=o S2 Jl. si ve 
we find the values of cos «, sin ¢, cos 2¢, sin 2e, ete. 
In case of the sine, as for example when 7 = 1, we have 
y—y i =2/ —I1 sine; butinze—2z1=2/ —1 sing, 
From the sec- 
we have the same factor, 2 »/—1, in the second member of the equation. 
From 
r= a(1—e cos €) 
we find 
7 
(=) = 1 — 2e cos ¢ + & cos 
e: = 1+ 2e cos « + 3¢ cos *e + 4¢ cos *e + ete. 
