140 
PROF. H. F. BAKER ON CERTAIN LINEAR 
an equation which does not contain sh/3 in its denominator. With a view to the 
comparison of this method with the two others given in the present paper we 
consider two examples. First, for the equation 
d 2 x 
dt 2 
+ [l + 4A& 1 w l 4" 4A k%w 2 -f-...J x — 0, 
for which n 2 is actually unity, we should determine [3 so that 
0 = k x chj3—^k 1 2 ch2/3 + k x ch/3 (2sh 2 /3—+k l k 2 ch(3+..., 
where we have replaced A by 1. This gives approximately 
ch/3 — —|A(l + %k 2 —k 2 ), sh/3 - i(l—^k 2 ), 
and hence 
q = ik x (l — ^-k 2 + k 2 + ...), 
while the value for j3, substituted for a, gives the series for x. We may remark that 
for the equation 
-y-f + (l + 8 k x cos 2 1) x = 0, 
C1/1/ 
Tisserand (‘ Bull. Astr.,’ IX., 1892, p. 102) finds 
2'. 3 
As a further example take 
d 2 x 
dt 2 
+ a;[l + 4^ 1 (l + ^4)+4Fw; 2 +...] = 0, 
which, as will appear, is an interesting equation. Then (3 is to be found from 
A = k 1 ch/3—^k 1 2 ch2^ + k 1 s ch/3(2sh 2 /3—^)+k 1 k 2 ch/3+..., 
so that 
and hence 
ch[3 — 1 +-gFj + — 3,k x k 2 +..., 
sA/3 = (*,)*()+¥*,-| + ■••), 
q = (k)HK-U‘-h+-)- 
In both these examples the value found for q follows at once from the general 
formula above given for q 2 , of which a further deduction is found below in Part II. 
In the last example the value found for /3 gives a solution for x in a series involving 
(A)"- It will be seen in Part II. that when x involves (A)% if is in a very simple 
way, and the case seems better treated as there explained. The occurrence of (A)* in 
