62 REPORT—1883. 
‘ His the earth’s mass. It has been so chosen that if the equilibrium theory 
of tides were fulfilled, with water covering the whole earth, the numerical 
factors in the ¥-12-Z functions would be each unity. The alterations of 
phase would also be zero, or, with land and sea as in reality, they might 
be computed by means of the five definite integrals involved in Sir 
William Thomson’s amended equilibrium theory of tides.’ 
The actual results of tidal analysis at any place are intended 
(see below, § 5) to be presented in a series of terms of the form 
fH cos (V+u—kx), where dV/dt or n, ‘the speed,’ is the rate of increase 
of the argument per unit time(say degrees per mean solar hour), and » 
is a constant. We require, therefore, to present all the terms of the 
X-%B-Z functions as cosines with a positive sign. When, therefore, in 
these functions we meet with a negative cosine we must change its sign 
and add x to the argument ; as the. X Z functions involve eines, we must 
add 47 to arguments of the negative sines, and subtract 47 from the 
arguments of the positive sines, and replace sines by cosines. The terms 
in the 4(X7?+ B?—-227) function require special consideration. The 
function of the latitude being }—sin?\, it follows that whenin the northern 
hemisphere it is high- water north of a certain critical latitude, it is low 
water on the opposite side of that parallel; and the same is true of the 
southern hemisphere. The critical latitude is that in which sin?A=4, or 
in Thomson’s amended equilibrium (!) theory, where sin?A=3(1+ 492). 
An approximate evaluation of $2, which depends on the distribution of 
land and sea, given in § 848 of the second edition of Thomson and 
Tait’s ‘ Natural Philosophy,’ shows that the critical latitudes are 35° N. 
and §. It will be best to adopt a uniform system for the whole earth, 
and to regard high-tide and high-water as consentaneous in the equa- 
torial belt, and of opposite meanings outside of the critical latitudes. In 
this Report we conceive the function always to be written +—sin?A, se 
that outside of the critical latitudes high-tide is low-water. Accordingly 
we must add z to the arguments of the negative cosines (if any) which 
occur in the function 3(.4?+ 3°-22Z). 
In continuing the development, the N-1}-Z functions will be written 
in the form appropriate to the equilibrium theory, with water covering 
the whole earth; for the actual case it is only necessary to multiply by 
the reducing factor, and to subtract the phase alteration x. As these are un- 
known constants for each place, they would only occur in the development 
as symbols of quantities to be deduced from observation. It will be under- 
stood, therefore, that in the following schedules ‘the argument’ is that 
part of the argument which is derived from theory, the true complete 
argument being ‘ the argument ’—«, where « is derived from observation. 
Following the plan suggested, and collecting results from (20), (26), 
(27), we have 
A= YP=(1— $e") p! c0s 2 (x—o,) + (1+ 4e4) 2p"y? cos 2x 
+iZep* cos (2x—3e Trae 
+thept/ {1—12 tan? 37 cos 2a,\cos (2y—0,-a7,—R+7) 
+ trent cos (2x —40,4+22,) 
+1%2mep* cos (2y—20,—s+2h—p) 
+}imep* cos (2y—20,+s—2h+p+r) 
+%3mp* cos (2x—20,—2s+2h) . 2. . . ww es 629) 
} Thomson and Tait’s Nat. Phil., or the Report on Tides for 1876. 
