208 
SIR G. H. DARWIN ON THE 
with a very high degree of accuracy. The differences of the tabulated numbers do 
not run with perfect smoothness, showing that there are residual errors of one or two 
units in the last place of decimals. The accuracy is however amply sufficient 
for the end in view, and it would have been wasteful to spend more time over 
the computations. 
§ 12. Determination of the Form of the Second Ellipsoid. 
The parameters r and K determine the form of the second ellipsoid in the same 
way that y and k determine the first. It is obvious that a jr is determinable in.two 
ways, and therefore any given value of y must correspond to a certain definite value 
of r. The fitting together of the two solutions can only be effected with accuracy 
by interpolation, but it would be so enormously laborious to find by mere conjecture 
the region in which to begin calculating with assumed values of T, that an approxi¬ 
mate solution of the problem becomes a practical necessity. 
After various trials T find that on neglecting e and r/ and writing 
sin 2 y 
X ~~ 7(3 + X + cos 2 y) 
the solution for k may be written approximately in the form 
K 
/2 
(1 + X) COS 2 y 
3 + X + cos 2 y 
[1—(9 —7X) X —(111 +228X —49X 2 ) x 2 
—rr(7875 + 28185X + 42333X 2 -3773X 3 ) x 3 ...] 
(41). 
If e were added to the numerator of the factor outside the bracket, and rj to the 
denominator, this formula would give nearly as good results as the more accurate 
method of the last section. 
Also I find 
a 3 - 4 (1 + X) cos B cos y sin 2 y r , q . . . , . 
^3 = —- 3 ^-t-cos 2 y - r [ 1 +!(5 + Mx + i(3055 + 17l8X+175X 2 ) x 
(3533389 + 3222607X+ 1021479X 2 +60025X 3 ) x 3 + ...J . . (42). 
In order to obtain the desired approximation, it is necessary to express a 3 /r 3 by a 
series which can be inverted; but this is not possible in the form just given, because 
cos (3 depends on k and therefore involves x . I find then by means of the above 
series for k' 2 that 
cos 0 cosy = l-f(7 + X) x + £(69 —106X—7X 2 ) x 2 + 1 * r (253 + 753X—1901X 2 —49X 3 ) X 3 ... 
