54 nrerorntT—1883. 
§ 2. Development of the Equilibrium Theory of Tides with reference 
to Tidal Observations. 
Tue first step is the formation of the tide-generating potential of the 
moon ; that for the sun may then be written down by symmetry. 
For this purpose we require to find certain spherical harmonic fune- 
tions of the moon’s coordinates, with reference to axes fixed in the 
earth. 
Let A, B,C (Fig. 1) be such axes, C being 
the north pole and AB the equator. 
Let X, Y, Z be a second set of axes, XY 
being the plane of the moon’s orbit. 
Let M be the projection of the moon in 
her orbit. 
Let [=ZC, the obliquity of the lunar orbit 
to the equator. 
Let y=AX=BCY. 
Let /=MX, the moon's longitude in her 
orbit, measured from X. 
Let 
M,=cos MA} ines : 
ae 1€ moon’s direction-cosines 
M,=cos MB with reference to ABC. - + Q@) 
M,=cos MC 
Then 
M,= coslcos y+sin/ sin x cos I 
M,=—coslsiny+sinlcosyxcosI;. . .'. . (2) 
M,==. smd sint 
We may observe that M, is derivable from M, by putting y+ 37 in 
place of x. 
Now for brevity let 
g—cos tT, g=sing 1)... a, ae 
Then (2) may be written 
M,= p? cos (x—/) +4? cos (x+/) 
M,=—yp’ sin (x- 1)—,? sin Loot » 2S ae 
M;= 2 pq sin l, 
Whence 
M?—M,2= p‘ cos 2(x—1) + 2p?q? cos 2x+q* cos 2(x +1) 
—2M,M, = the same with sines in place of cosines. 
M,Mz = —p*q cos (x—21) +pq (p?—9") cos x + pq? cos (x +21) (5) 
M,M, = the same with sines in place of cosines. 
g—My~= § (pt—sp°P +4) +279? cos 21 
These are the required spherical harmonic functions of 1, M,, M3. 
Let M denote the projection of the moon on the celestial sphere con- 
centric with the earth, and P that of any other point. 
Let 7, p be the radius- vectors of the moon and of P respectively, and ~ 
let &, n, £ be the direction-cosines of P, with reference to the axes A, B, C. 
