ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES- 
229 
and with this value for we find 
= - -008529, 
L ^2 = — -013306, 
= - -016251. 
The value of log- -- 7 ^—-rr is 
® (2?i + 1) {2n + 3)2 {2n + 0 ) 
L 3 = -27916, 
= -064345, 
Lg = -017624, 
for n = 2, 4-6566, for n — 8 , 6-8898, 
„ n = 4, 5-8490, „ n - 10, 6-5564, 
„ 71 = 6 , 5-3034, „ n = 12, 6-2769. 
With these values we find for the successive convergeots to the continued frac¬ 
tion Hg 
-001141, -001217, -001219, 
while the successive convergents to the continued fraction 
- -000910, - -000939, - -000940, . . . 
It will be observed that these continued fractions converge with great rapidity; so 
long as the depth of the ocean is not less than that we are here using, I find that 
when X^doj® has a value in the neighbourhood of a root of the equation = 0 , the 
continued fractions H„_ 2 , K «+2 are represented without sensible error by their fourth 
convergents, while in many cases the second convergents will form a sufiicientl}'’ 
accurate approximation to their values; this rapid convergence of course greatly 
facilitates the numerical computation. In practice, the simplest method of evaluating 
the continued fractions is to assume that, for a sufficiently large value of n, K„ = 0 , 
and then to compute K,;_ 3 , &c., in succession from the formula 
K„_o = 
{271 - 7) {'2n - 5)2 {2n - 3) 
_ 2 IV,; 
Thus, in the present instance, we may put K^g = 0 , and deduce 
log Ki 4 = ^4*06, log K ^3 = n4-436, log K^q = u.4'9730. 
K 
]o — 
whence, as above, 
- -000940. 
