WITH THE TIDES OF A YISCOUS SPHEROID. 
583 
Since the equilibrium tide, due to the disturbing potential, would be given by 
yi-l 
<T; = 
2(^—1) 9 
2i + 1 a 
Q; cos vt. 
(27+1) a 0 
it follows that inertia augments the height of tide in the proportion 1 : 1 —- —— — v z . 
In the case where i— 2, the augmentation is in the proportion 1 : 1 — | 
We will now consider the nature of the motion by which each particle assumes its 
successive positions. 
With the value of cr, given in (76) 
o 2(7-1) g _ 
/-v • . -« . 
27 + 1 a} 
g 
cos vt. 
Then substituting in (74) 
da_ 
clt 
(27 +1) a 
d v* cos vt Q o’ 1 
dx 27(7—1) g 3 
— ir 
1 
2 ^ ■{“ 1 CL ( 
and two similar equations j 
Integrating with regard to t 
(77) 
d Qi'r' v sin vt 
dx 27(7—1) g 
— V“ 
2 7 +1 a 
and two similar equations j 
!- 
• (78) 
There might be a term introduced by integration, independent of the time, but this 
term must be zero, because if there were no disturbing force there would be no flow. 
Hence it is clear that there is a velocity potential d, and that 
3 = 
27(7—1) g dt 
27 +1 a 
i<y S-) 
(79) 
Now however slowly the motion takes place, there will always be a velocity potential, 
and if it be slow enough we may omit v 3 in the denominator of (79). In other words, 
if inertia be neglected the velocity potential is 
2z + l a cl. . 
' 2i(i-l)gdt'^‘ 
4 F 
MDCOCLXXIX. 
