ANALYSIS TO THE DYNAMICAL THEORY OP THE TIDES. 
223 
\-u, 
0, 
0, 
1 sV 
— Ijj, 
^4> 
0, 
. 
1 0, 
^4. 
big. 
’7c 
- 
’7r—65 
0 
f.-e, 
L ,._ 4 , 
Vr-i 
. 
0, 
e-4, 
- L ,._2 
It follows that there exist certain types of free oscillation for which all the values 
of C with sujEhxes greater than r are zero. For these types the height of the surface- 
waves will be expressible by a finite series of terms terminating with a term 
involving P,.. 
In like manner, if the disturbing force be derivable from a potential function of the 
form of a second order harmonic, the equations which determine the forced 
oscillations are 
'^■ 2^-2 d" . V-) “h 
7 4a)2«2’ 
^4^4 ~ + ’76^8 = 
^-A -2 
L,._20,._2 - 0, 
L,.C,. -p 'q,Gr+2 — 0) 
L, +2Ci.+2 'P Vr + 2^r+i — 0* 
If we suppose C,.^ 2 ) ^r+i, • • • all zero, the first r/2 of these equations will serve to 
determine Cg, C^, . . . C,- in terms of y^) while the remaining equations will be 
satisfied identically. Thus the forced tides for the law of depth in question will be 
expressible by a finite series of terms terminating with a term involving P,.. 
This general law does not hold when r = 2, owing to the presence of a term in the 
right-hand member of the second equation. 
The fact that for these laws of depth the tide-heights could be expressed by finite 
series, instead of by the infinite series usually required, was originally proved by 
Laplace in the ‘ Mecanique Celeste.’'^' 
* Part I., Book IV,, § 5. 
