Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

158 MESSRS GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY
On eliminating b from (A) and (B) we obtain an equation for X2.
Now we have seen that 4 sin2 b/2 is of the order S'L'/EL, where E stands for the
whole capacity. In the most important cases S, the external capacity, is large compared
with S1, the capacity of the coil; and in this case the whole capacity is large compared
with Sl. In the general case on substituting in (B) from (A) we find
x2L' {sin (n - 2) q - 2 sin (n - 1) + 2 sin (n - 1) (1 - cos )}
= s {sin (n - 2) s - sin (n - 1) s + 2 (1 - cos ) sin ( - 1) 6}.
Whence 2L'S = 1 + - (n - 1)
sin (n - 2)p - 2 cos 5 sin (n -1)b
-= 1sin(n- 1)
sin nb
= 1 - cos + sin r cot nb,
2L' (S - S') = sin cot n..............................(C),
or substituting for X2L' from (A)
2 tan (S - S') = S' cot n....................... (C'),
the fundamental equation for the periods.
Up to the present no assumption has been made as to the relative values of S
and S'. In our case S' is small compared with S1, and then o is small; S may have
any value.
In the more important cases S is large compared with S1. In this case ne) is of the
order (Si/S)L. We may expand cotnas and cot ~/2 and use 1/SL as an approximate value
for X2 in the small terns.
Now we have 2 (1 -cos s) = XL'S'.
Thus s2 = 2L'S' (1 + -2X2L'S' +...).
Hence 2 = X2S {l + + 4+...
=' 1 n+ + +"
where a, b, etc. can be found approximately. Hence expanding in Bernouilli's numbers,
28 2 / +4 *
1 P- L4...*
22B,02 B244 1
Now, as a first approximation,
'n2L'' n S, S ( 1
LS n -1S S \ n)