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MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
Then clearly 
V- 
(1 —e 2 ) 
^=2^XY+2 
e-v* x 2 —y : 
■fSfy£YZ+2&XZ+t 
+ X 2 + Y 2 —2Z 2 
Now assume that the five functions 2XY, X 3 —Y 3 , YZ, XZ, X 3 +Y 3 —2Z 3 are each 
expressed as a series of simple time-harmonics ; it will appear below that this may 
always be done. We now have V expressed as the sum of five solid harmonics p~tp, 
p 2 (£ 2 —p 3 ), &c., each multiplied by a simple time-harmonic. According to any tidal 
theory each such term must raise a tide expressible by a surface harmonic of the 
same type, and multiplied by a simple time-harmonic of the same speed ; moreover, 
each such tide must have a height which is some fraction of the corresponding equi¬ 
librium tide of a perfectly fluid spheroid, but the simple time-harmonic will in general 
be altered in phase. 
Now if r=a-\-a- be the equation to the wave-surface, Corresponding to a gene¬ 
rating potential Y=[r/(1 — e 3 ) 3 ] p 3 2fpXY, then when the spheroid is perfectly fluid, 
a 'a=[r g(l—e 3 ) 3 ] 2^pXY, where g=fc//a, according to the ordinary equilibrium 
theory of tides. (It will now be assumed that we are dealing with bodily tides of the 
spheroid ; if the tides were oceanic a slight modification would have to be introduced.) 
In a frictional fluid, the tide cr will be reduced in height and altered in phase. 
Let ^ represent a function of the same form as XY, save that each simple time- 
harmonic term of XY is multiplied by some fraction expressive of reduction of height 
of tide, and that the argument of each such simple harmonic term is altered in phase; 
the constants so introduced will be functions of the constitution of the spheroid, and 
of the speed of the harmonic terms. Also extend the same notation to the other 
functions of X, Y, Z which occur in Y. 
Then it is clear that, if r=ci-\-cr be the equation to the complete wave surface 
corresponding to the potential Y,. 
( 1 -e 2 ) 3 ? ?=2f, vj?+U .y *tjr+2ij{ m +m m 
t a 
+ 2 3 3 • W 
This expression shows that cr is a surface harmonic of the second order. 
Then by (17) we have for the disturbing function for the moon, due to Diana’s tides, 
where cr is the height of tide, at the point where the moon’s radius vector pierces the 
wave surface. 
