DIFFERENTIAL EQUATIONS OF ASTRONOMICAL INTEREST. 
139 
If from these formulse we determine M x and M 2 — in terms of h 2 and h 3 we 
find for g, 
q = k v sha\ + Ho ctha . A 2 
3 r H 2 3 2kjl 2 ch 2 a h 3 cha — k- i k 2 —^k 2 (2sh 2 a — l)~\ 
L 2 k lS h 3 a sha sha _| + *"’ 
where 
H 2 = h 2 —bk 2 (2sh 2 a—l). 
This formulae is apparently unsatisfactory when sha is small, or n 2 — 1 nearly equal 
to 4A/q. In fact, the series is of the form 
a+ — A + 
2 a 
±ca 2 -b 2 
8 a 3 
2 , Sa*d— ici 2 bc + b 3 3 
A + 
16a 5 
A 5 + 
whose square has a form in which we can put a = 0, On squaring, we have 
q 2 = (h 2 -k 2 )\ 2 +2h 1 K 2 \ 3 +\ i (K 2 2 -±h 2 K 2 +2h 1 h s -2k 2 k 2 -h 2 k 2 +W)+.-., 
wherein 
Ho = h 2 -h 2 + W, 
and this form is appropriate when a = 0 or h Y — k x . In particular, when 
h 2 = h 3 — ... = 0, but h 1 is not zero, this gives 
T 
q 2 = {h 2 -k 2 ) \ 2 +h 1 {3k 2 -2h 1 2 )\ 3 +[5 {h 2 -k 2 ) 2 - - 2k 2 k^ A 4 +..., 
a formula reproducing the former if /q + h 2 X + h 3 X 2 be put for h x . It will be seen in 
Part II. of this paper why the form of q 2 is comparatively so simple. 
Brief reference may be made to another way in which we may use the foregoing 
equations, regarding h u h 2 , h 3 , ... not as given constants but as quantities to be 
determined to simplify the result; this has been adopted by Prof. W hittaker 
(‘ Proc. Math. Soc.,’ Edinburgh, XXXII., 1913-14) who chooses as his condition that 
no terms in £°, f 1 shall occur in W u W 2 , ..., in the expression for x. This can be 
secured by taking 
P x = k x sha, Mj = 2k x sha, P 2 = 0, Q 2 = 0, .... 
From our present point of view a more natural procedure is to take 
Pi = 0 = Q x = P 2 = Q 2 =_ Then we obtain 
n 2 — l+Xk 1 ch/3—^\ 2 k 1 2 ch2^+\ 3 [k 1 3 ch^(2sh 2 ^—^) + k 1 k 2 chl3]+..., 
where we have written /3 in place of a, as this argument is now supposed to be 
determined, from this equation, corresponding to a given value of n 2 . When (3 is so 
determined, q is given by 
q = k^Xshfi — k 2 X 2 sh2(3 + A 3 [Jc x shf3 (6ch 2 /3—hr) + k^sh/3] + ..., 
u 2 
