84 REPORT— -1883. 
Then © 
cos I=cos 7 sec 6 cos (w+/3) 
sin y=sin 7 cosec I sin N (55) 
sin (N—£)=sin w cosec J sin N 
The formule (53) also lead to the rigorous formule 
- sini cot sin N (1—tan 47 tan w cos N) 
tan ¢==—_— Se ON 
cos* 41+sin 7 cot w cos N—sin? $i cos 2N ; 
; (&3’) 
tan 7 cosec w sin N 
tan oS 
1+tan 7 cot w cos N J 
But, if we treat 7 as small, (53’) may be reduced to 
ele : PAE 1—4 sin? w 
tan 6=7 cot w sin N—1i? sin 2N —2 
Ff sin? w 
FOr! 
; ; eae + COS w » (538) 
tan =7 cosec w sin N—}1? sin 2N —— | ( 
i si” w 
cos I=(1—4:”) cos w—i sin w cos N 
A table of values of &, », I, for different values of N, with w=28° 273, 
i=5° 8/8, may be computed either directly from (53) or from (55). 
We give below in § 12 a table for J, », & for every 2° of N, computed 
from (55) under the superintendence of Major Baird, at Poona. 
The approximate formule (53’’) will be of service hereafter. 
On the Mean Values of the Coefficients in Schedules [B.]. 
In the three schedules [ B] of lunar tides, ‘ the coefficients’ are certain 
functions of 7, and there are certain terms in the arguments which are 
functions of y and £. We may typify all the terms by J cos (7+), where 
J isafunction of J,andwof yvand &. If we substitute for J and w in terms 
of w, 7, N, and develop the result, we shall obtain a series of terms of 
which the one independent of N is, say,J, cos JT. Then J, is the mean 
value of the semi-range of the tide in question. Such a development 
may be carried out rigorously, but it involves a good deal of analysis to 
do so; we shall therefore confine ourselves to an approximate treatment 
of the question, using the formule (53’’) for ¢ and v. 
It may be proved that in no case does J involve a term with a sine of 
an odd multiple of N, and the formule (54) or (55) show that in every 
term of sin wu there will occur a sine of an odd multiple of N; whence it 
follows that J sin uw has mean value zero, and J, is the term independent 
of N in J cos wu. ; 
It may also be proved that in no case does cos u involve a term in ccs N, 
and that the terms in cos 2N are all of order 7”; also it appears that J 
always involves a term in cos N, and also terms in cos 2N of order 72. 
Hence to the degree of approximation adopted, J, is equal to J, cos u,, 
where J, is the mean value of J, and cos uw, the mean value of cos wu. 
In evaluating cos u, from the formule (53), we may observe that 
wherever sin? N occurs it may be replaced by §; for sin? N=}—4 cos 2N, 
and the cos 2N has mean value zero. 
