86 REPORT— 1883. 
If we write 
Z=Ccos $w cos }i—sin }w sin }¢ e% 
c=sin 4 cos 4/+ cos dw sin di e™ 
where « stands for ./—1; and let =,, «, denote the same functions with 
the sign of N changed, then it may be proved rigorously that 
cos! $I cos 2(v—£)=}(at+ a) 
sin? J cos 2v=2(a7«)?+4+ a 7x?) 
sin I cos? 4I cos (2§— 1) =m «+ a 3K, 
sin I sin? iI cos (264+71)=a3+a,K,3 
sin I cos I cos v=(@x, +k) (wa, —kk)) 
sin? I cos 2£=2( ax? + w 7x7) 
1—$ sin? T=a@?a\?—4a00 KK, +46)? 
The proof of these formulx, and the subsequent development of the 
fanctions of the a’s and x’s, constitute the rigorous proof of the formule, 
of which the approximate proof has been indicated above. The analogy 
between the a’s and «’s, and the p, g of the earlier developments of this 
Report, is that if 7 vanishes c=a,=p, «=«\ =. 
[See a paper in the Phil. Trans. R.S. Part II. 1880, p. 713.] 
This investigation justifies the statements preceding the schedules 
[B] as to the mean values of the coefficients. 
Formule for computing f. 
In the original reduction of tidal observations we want 1/f; in the 
use of the tide-predicter f is required. 
On looking through the schedules [B. ], we see that the following values 
of 1/f are required. 
(1) cost Sw cos! $7 (2) sin? w (1—# sin?) (3) sin w. cos? Sw cos! $7 
a a Sa ae - a Te Sy | ae Ll . 72s 
cost3f ’ sin? 1 ; sin I cos?32 0’ 
(4) sin w sin? dw cost $7 (5) sin w cos w (1— sin? 7) 
sin isin? $f” sin 1 cos I 
? 
sin? w cost hi... (1—3 sin? w) (1—3 sin? 2) 
©) ae >?) SO geata® Dp 
And in the case of the over-tides and compound tides (schedules 
| F], [H]), powers and products of these quantities. 
A table of values of these functions for various values of I is given in 
§ 12. , 
The functions (2) and (5) are required for computing f for the K, 
and K, tides. 
In this list of functions let us call that numbered (2) &., and that 
numbered (5) k,; %, and &, being the values of the reciprocal of f which 
would have to be applied in the cases of the K, and K, tides, if the sun 
did not exist. 
