14 
SIR G. H. DARWIN: FURTHER CONSIDERATION OF THE 
the correct values are given in the table below. I find that for <H 4 quadratures gave 
too small a value by a 3-0 0 th part; for <H 6 by a y^th part; for Hs by an -/yth part. 
The method which I have given above fails for the tenth zonal harmonic, unless 
we use logarithms of more than seven places; and it is not worth while to 
undertake so heavy a piece of computation. I conclude by extrapolation that for ®l 10 
quadrature (carried out on exactly the same plan as in all the other cases) gives too 
small a result by a - 7 - 0 -th part of itself. I therefore augment in this case the result of 
the quadratures and find ^ 10 = 0 - l 1 640 ; this enables us also to compute Bn> 
The following table gives the results of the whole computation* :— 
Table of Logarithms of <H/ and 13/ Integrals. 
i. s. 
log ‘H ( : s + 10. 
log BA 
2 0 
9•6931231 
•0929494 
2 2 
9•3330037 
•4066504 
3 0 
9-54617 
•20462 
4 0 
9-4332383 
•2657402 
4 2 
9-24250 
•39502 
4 4 
9-04753 
•43121 
6 0 
9-2701270 
•3263106 
6 2 
9-14462 
•39512 
6 4 
9-00632 
• 42458 
8 0 
9•15835 
• 35745 
10 0 
9-06595 
•36897 
§ 3. Note on § 15. “ The Determination of certain Integrals .” 
It has been found best to make some changes in this part of the work. 
The integrals to be evaluated are denoted 
The integral 2p A 2m 2n may be made to depend on °A 2/Ji 2 ” (which is the same as A 2m 2n of 
the original paper), and therefore I only evaluate the latter. 
These integrals were originally found as the differences of certain other functions, 
but it is not hard to give formulae for finding them directly. I have done this and 
recomputed the whole series of values, i 
* The only error of any moment which I have found in my previous work is that in some of the cases 
I had forgotten to introduce the factor k in some of the H’s after effecting the quadratures. The error in 
my results from this oversight was fortunately not serious, 
f There is a misprint on p. 286. The function U 6 ° should be 
OTT 
16 cos /3 cos y 
[sec 4 ft + sec 4 y + 
sec 2 /3 sec 2 7 ]. 
