WITH THE TIDES OF A VISCOUS SPHEROID. 
577 
each of which corresponded to one of the constituent simple harmonic tides. Each 
such term involved two factors, one of which was the height of the tide, and the other 
the sine of the lag. Now if e be the lag and v the speed of the tide, it was found in 
the Erst approximation that tan e=19uy-E2 gaw, and that the height of tide was pro¬ 
portional to cos e ; hence each term had a factor sin 2e. 
But from the present investigation it appears that, with the same value of e, the 
( ^2 
1 r 5 Q~ cos 3 e 
whilst the lag is 
e H~T 5 Vg sin e cos e, so that its sine is (1 + T^o~ cos " e ) sin e - 
Hence in place of sin 2e, we ought to have put sin 2ef 1 —Hi 
7 9 ‘ 
5 0 
COS“ e 
or 
m a 
sin 2 e^l cos 3 ej. 
Thus every term in the expressions for , — should be augmented, each 
J 1 dt* dt ’ clt & 
proportion depending on the speed and lag of the tide from which it takes its origin. 
In the paper on “ Precession,” two numerical integrations were given of the 
differential equations for the secular changes in the variables; in the first of these, 
in Section 15, the viscosity was not supposed to be small, and was constant, in the 
second, in Section 17, it was merely supposed that the alteration of phase of each tide 
was small, and the viscosity was left indeterminate. It is not proposed to determine 
directly the correction to the first solution. 
The correcting factor for the expression sin 2 e is greatest when e is small, because 
cos 3 e may then be replaced in it by unity; hence the correction in the second 
integration will necessarily be larger than in the first, and a superior limit to the 
correction to the first integration may be found. 
We have tides of the seven speeds 2 (n—/ 2 ), 2 n, 2(n-\-f2), n — 2 / 2 , n, n-\-2fl, 2 / 2 ; 
hence if the viscosity be small, the correcting factors for the expressions sin 4e l5 sin 4e, 
sin 4Co, sin 2e\, sin 2e, sin 2eb, sin 4e" are respectively 1 + multiplied by the 
squares of the above seven speeds. 
Then if \=—, the seven factors may be written 
1 -pVlAfi multiplied by (1—\) 3 , 1, (1+X) 3 , for semi-diurnal terms 
1+y-f-— multiplied by (1— 2\)~, 1, (1+2X) 3 , for diurnal terms 
2 
and 1 + fi-fi 6 -—X 3 , for the fortnightly term 
(63) 
* The following method of correcting the woidc of the paper on “ Precession” has been rewritten, and 
was inserted on the 17th May, 1879. 
4 E 2 
