306 
ME. GEORGE H. DARWIN ON THE INFLUENCE OE 
and since ^ is zero, when <p= + ^, for all values of t, §= — ~t cos 2 <p sin 2 0 , and 
d§ 2h 0 . . 
-j7 = — - cos 2 <p sin 0 . 
at c r 
Hence II 2 , twice the area conserved on the plane of xz, is 
ffi 
’ 9 * /I 7 7 a 7 
sm 0 ar «0 « p . cos <p, 
taken from r= 0 to c, 0=0 to tt, <p = — - to +^. 
If the sphere be taken as homogeneous, 
£,= ■ 
24 
c % 
r 4 sin 3 0 cos 2 <p cos <p dr db d<p 
y/ 2 Jipc*= MAc, 
45 V s 15tt ’ 
and // 3 are both clearly zero. 
The above value of H 2 is larger than what it would be in the case of the earth, if 
Laplace’s law of internal density were true, because the external layers have been taken 
too heavy, and the internal too light. But taking that law of density, A=^Mc 2 very 
nearly. 
Hence 
A 5 it c 
If we let the time run on until the highest point of the continent has risen one foot, 
so that — =- , 
c 20,900,000 
then — 2 ‘ = — — aA. 
A 5?r 20,900,000 
But reference to sec. 21 (fig. 3) shows that * v, = -5480/i, or in the present notation, 
0*= ' 5X 64^000 nearl y- 
Therefore 
&=- 8x 648 V I J— nearl 
A/3 104,500tt 2 141 J 
But generally, since the angular velocities a, /3, 7 of the moving axes, to which 
lb, refer, are very small, therefore 
»,=#, 
to the first order of small quantities, within the limited period to which the investigation 
applies. So that in this particular case, 
||= — nearly, and © 1 =^ 3 = 0 . 
And, besides, this value of ^ is larger than it ought to be, because was calculated 
on an assumed homogeneity of the earth. This, then, justifies the conclusion in the 
text on p. 279. 
