64 REPORT—1883. 
Since ¢,— a, is the moon’s mean anomaly, we have 
s—p = 6-2, 
Let p’ be the longitude of the perigee, measured from Y in the ecliptic. 
lf P in Fig. 2 be perigee, we have by the ordinary formula for reduc- 
tion to the ecliptic, 
2P=p'—N+j sin? é sin 2 (p’—N) 
But @ =IP=18 + 8P=VT&—£+ &P 
=p’—f+1 sin? 7 sin 2 (p’—N) 
Now p=p' +i sin? i sin 2 (p'—N), and therefore 
=. 
Peake igciaeeF rere eres a 
Again 
Sais A.T — 1 =e—y veeeenh ooheeren be (33) 
In this formula we suppose g to increase uniformly from the time 
when the tidal observations begin. 
Since in all the tidal observations local mean solar time is used, it 
will be better to substitute for g in terms of local mean solar time 
and the sun’s mean longitude. Let ¢ be local mean solar time reduced to 
angle, so that at noon t=0°. Let h be the sun’s mean longitude ; here- 
after we shall write p, for the longitude of the sun’s perigee. 
Then we have 
ytth—y. ope, & oh Sy be ee (34) 
We shall now substitute from (32) and (34) in the Y-Y-Z functions 
(29), (80), (81) ; substitute from them in (28), and express the final result 
in the form of three schedules (pp. 18, 19, 20). 
The schedules are arranged thus, First, there is the general coefficient 
3 
2 ai) a which multiplies every term of all the schedules. Secondly, 
there are general coefficients one for each schedule, viz. cos? for the 
iy as 
semi-diurnal terms, sin 2 for the diurnal, and }—# sin?A for the terms 
of long period. These three functions of the latitude of the place of 
observation are the values at that place of three surface spherical har- 
monic fanctions, which functions have the maximum value unity, at the 
equator for the semi-diurnal, in latitude 45° for the diurnal, and at the 
pole for the terms of long period. 
First, in each schedule there is a column of coefficients, fanctions of 
T and e (and in two cases also of p). 
In the second column is given the mean semi-range of the correspond- 
ing term. This is approximately the value of the coefficient in the. first 
column when J=w. We forestall results given below so far as to state 
that the mean value is to be found by putting [=w in the ‘ coefficient,’ 
and when the function of I is cos ‘447, sin Icos *4J, sin Isin*43J, sin ?J (in 
B, iii.) multiplying further by cos *57; and where the function of J is sin ?I 
(in B, i.) sin Tos I, 1—$ sin? multiplying by 1—} sin *i. 
Thirdly, there is a column of arguments, linear functions of t, h, s, 
