22 A NEW METHOD OF DETERMINING 
Using the values of C, q, @, Q, just found, Hint, in his New Theory of Jupiter 
and Saturn, has given another expression for ) which we shall employ. 
To transform 
(2): = (C—q. cos (¢ — Q)) (L—m. cos (¢ + Q)) 
into the required form we put 
C= 1B 
= sin x, = sin 7% 
Gi 
a=t93%, b=tWan (27) 
_ seC 2%. 8eC2 % 
NORE 
Then 
a) = O[1—sin x . cos (¢ — @Q) | [1—sin y.. cos (e+ @Q) | 
_ C[ sec? by (1—sin x . cos (e’ — @)) | [sec? $41 (1 — sin x: . eos (¢’ + Q)) | 
Ce, sec” by see’ 3% 
_ Of 1+ ty? by — 2ty 4 cos (e’ — Q) | [1 + ty? by, —2lg yn cos (e’ + Q) | 
ta “sec” by sec’ 2Y, 
Substituting the values of a, 6, NV, we get 
(2)" = _W* [1 + @— 2a cos (¢— Q) | * [1+ — B cos’ + Q]? (28) 
We compute the values of a, b, NV, corresponding to the different values of g, and 
check by finding the sums of the odd and the even orders, which should be nearly the 
same. If we put 
[1 + a — 2a cos (¢ — Or = [4 6° + 6. cos 0+ 6 . cos 29 + 6°. cos 39 + ete. | 
[1 + 0 —2b cos (e + ®) | = [3 BO + B .cos(¢+ Q)+ B”. cos 2 (e+ Q) + ete. | 
where s = ”, = ¢ — Q, we are enabled to make use of coefficients already known. 
9? 
