230 
MR. S. S. HOUGH ON THE APPLICACTION OP HARMONIC 
In like manner, in order to evaluate H„_ 2 , we assume that, for some sufficiently 
small value of r, H, = 0, and then compute H. + o, ^r + i, • . . = H „_3 in turn by 
means of the formula 
1 
Tr (2r + 5) (2r + 7)^ (2r + 9) 
xl,- + 2 — • 
4- 2 XX;. 
In the present case we find 
log Hg = 3-2107, log = 3'0516, log Hg = 3-0860, 
whence 
Hg = -001219, 
and 
Lg - Hg - K^o = - -001219 + -000940 
= — -000279. 
This being a small quantity we conclude that there is a root of the period-equation 
differing but slightly from the value assumed for A.^/4cu^, namely, 2-23726. A closer 
approximation will be found by using this value in Hg, K^g, and again equating 
Lg — Hg — Kjq to zero; in other words, by putting 
Lg = + -000279. 
The second approximation to the root is therefore given by 
X74a;2 = 2-25735. 
Taking this value, and proceeding as before, we find 
Lg - Hg — Kjo = -000279 — -001183 + -000961 
= -000057. 
We have now found that, when — 2’23726, 
Lg - Hg — Kjo = - -000279, 
and when = 2-25735, 
Lg - Hg - Kjo = + -000057, 
whence, by interpolation, we conclude that 
Lg Hg = 0, 
when 
X3/4w" = 2-25394. 
In general we shall at this stage obtain a sufficiently close approximation to the 
root sought, as may be verified by actual substitution. Should however great 
accuracy be desired, we may re-start the computation, using the value already found 
as a first approximation, and so continue until the desired degree of accuracy is 
attained. The number of cases in which I have found a repetition of the process 
necessary is however extremely limited. 
