AND ON THE REMOTE HISTORY OF THE EARTH. 
453 
Then equating coefficients in these two expressions (o') and (G) 
Whence 
as 2 —i= — T+iX+1 — C,pY) 
f~i= <mX+i(l —GpV) 
* 2 -4= -fX-i(l-6pV) 
f-^= ffi+x+^i-G^Y)- 
z 2 —»r= d>—X—1(1 — 6pV) 
a? 3 —y 2 = — 2<f> 
also }> 
j,~—I'?' 
—^T r 
»*/= “ T' 
(7) 
These six equations (7) are the desired functions of x, y, z in terms of the auxiliary 
functions. 
§ 2. The form of the spheroid as tidally distorted. 
The tide-generating potential has thirteen terms, each consisting of a solid harmonic 
of the second degree multiplied by a simple harmonic function of the time, viz. : three 
in <I>, three in <f>', three in T - , three in T' 1 ', and one in X. The fourteenth term of Y 
can raise no proper tide, because it is independent of the time, but it produces a 
permanent increment to the ellipticity of the mean spheroid. 
Hence according to our hypothesis, explained in the introductory remarks, there 
will be thirteen distinct simple tides; the three tides corresponding to <f>' may 
however be compounded with the three in <E>, and similarly the T' tides with the 
' V F tides. Hence there are seven tides with speeds* [2n—2S2, 2 n, 2 ft+ 2/2], [ft— 2/2, 
n, [2 12], and each of these will be retarded by its own special amount. 
The ( l> tides have periods of nearly a half-day, and will be called the slow, sidereal, 
and fast semi-diurnal tides, the tides have periods of nearly a day, and will be called 
the slow, sidereal, and fast diurnal tides, and the X tide has a period of a fortnight, 
and is called the fortnightly tide. 
The retardation of phase of each tide will be called the “lag,” and the height of 
each tide will be expressed as a fraction of the corresponding equilibrium tide of a 
perfectly fluid spheroid. Then the following schedule gives the symbols to be 
introduced to express lag and reduction of tide 
* The useful term “ speed ” is due, I believe, to Sir William Thomson, and is much wanted to indicate the 
angular velocity of the radius of a circle, the inclination of which to a fixed radius gives the argument of 
a trigonometrical term. It will be used throughout this paper to indicate v, as it occurs in expressions of 
the type cos (vt+ >j). 
