222 
SIR G. H. DARWIN ON THE 
of quadrature only to the differences u n -v n . The result is a correction 
[ Vdxl>. 
Jo 
The empirical function which satisfies these conditions is 
to the integral 
When xfj = y = 108, v = u 10 ; and when xjj = 98, v - u 10 e '° ge ^ = u a . 
Then [vchh = f -^- ° S , (i- e -mog,(«,„/«»)) 
Jo lo ge{u m /u 9 ) x r 
In the cases we have to treat e ~ w}0 ^^) is an extremely small fraction, so that 
practically v (1\[j — - ^ ’ au( ^ * s ^ ie f llnc ti° n to be corrected by the 
result of quadrature. 
For the quadratures we have 
^10 = u 10 , V 9 = u 9 , v s = u 10 f^-V, v 7 = u 10 ( ^Y, &c. 
Wio/ V^io/ 
Thus the equidistant values of the function to be integrated (arranged backwards) 
are 
0, 0, u s \ , u 7 —u 10 (—\ , &c. 
V^IO/ V^lO / 
The first two are zero, the next three or four are sensible, and the rest are 
insensible; thus the quadrature is very short. The correction is found to be very 
small, and we might perhaps have been content with the empirical integral without 
material loss of accuracy. 
17. The Fifth Zonal Harmonic Inequality. 
this is treated exactly in the same way as the fourth, and I will only oive the 
results. & 
We have 
r 
k 
% ( v ) 
C 5 M 
3T 5 
- 7r<r T T £(1 + H p) (1 - 5A + WA a ), 
A M HA A» +rkA 2 A‘ (v), 
\/(l-^ 0 cos 2<£).[l-7/3 u cos 2</> + f|/3 0 
cos 
— 1 
Whence 
