HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 35 
Then pé, pn, pf are the coordinates of P, and rM,, rM,, rM, those of M. 
If M be the moon’s mass, and u the attraction between unit masses at 
unit distance apart, then by the usual theory the tide-generating potential 
V, due to the moon, of the second order of harmonics, at the point P, is 
given by 
V=g EX pt (co PM—2) (6) 
But since cos PM=i VW, +7 M,4+2 Ms, 
> f2—n? M2 Me? 
cos? PM—2=2'n M,M,4+2 a = : 
pat O72 242. OV,2 
43 ae 22 oe M, bs hotknieer HACE) 
Now let c be the moon’s mean distance, e the eccentricity of the 
moon’s orbit, and let 
4+2nfM,M;+22¢ MM, 
tT 
3M M 
2 48 
2 (8) 
C 
Then putting 
xafC= Pan, vale Cr an, za[2C= Pa, 
i ele “ 
We have 
pe 8 9tn KV 42 ial AOL" 4 nf YZ 4220 X7 
(1—e?)3 2 2 : 
z24 2) 972! x2 29772 
+3737 vals Be aa OteeLD 
A simple tide may be defined as a spherical harmonic deformation of 
the waters of the ocean which executes a simple harmonic motion in time. 
Corresponding to this definition the expression for each term of the tide- 
generating potential should consist of a solid spherical harmonic, multi- 
plied by a simple time-harmonic. 
In (10) p%én, p?(é?—n?), &e., are solid spherical harmonics, and in 
order to complete the expression for V it is necessary to develop the five 
functions of X, Y, Z in a series of simple time-harmonics. 
It will be now convenient to introduce certain auxiliary functions, 
namely 
® (@=[2C=! cos (21-42), 
¥ ()=[- C=] cos a, El 
/ 
(11) 
Then from (5) and (9) we have 
M2 V2 po (—2y) +2p%4?¥ (2x) +44 (2x) 
2XY= the same with x +n for x. 
Y¥Z=— p*q® (—x) + pg (p?—9) ¥ (x) tPF (x) ~ (12) 
XZ= the same with x—47 for y. 
3 (RP+Y?—227)= 5 (pt —4p?q? +44) R + 2p2q? (0) 
