204 
MR. HOPKINS'S RESEARCHES IN PHYSICAL GEOLOGY. 
To find the value of the second term of the denominator in the above value of r, I 
shall take the well-known expression for the density, viz. 
s' — A 
where q' a x = 150°. This gives us 
sin q 1 a 1 
/ ( a A . 
a! 2 da! — (sin q 1 a — q' a cos q' a) ; 
whence 
Also 
A 
- 2 (sin q' a- q 1 a cos q' a) = g' — • 
^ a ! 4 da! = ^4-^4 ^ (6 q 1 a — q' 3 a 3 ) cos q' a + (3 q' 2 a 2 — 6) sin q' a j- ; 
and substituting the value of A from the preceding equation, 
4 7 r , , , (6 g' a — q VA a 3 ) c os q 1 a + (3 g' 2 a 2 — 6) sin g' a , 
(PJ 0 ^ a a g' 2 a 2 (sin g' a — g' a cos g' a) & a ’ 
The numerical value of the multiplier of g' a in this expression is nearly constant for 
3 . 1 
different values of a, from a = 0 to a = a x , being in the former case and (very 
nearly) in the latter. If we take the mean of these values for that of the quantity in 
question, 
J o £ a 4 d a —^g a very nearly. 
Hence we have 
r = a{l+f ; 
and if g be the value of g' at the earth’s surface where a — a x 
and therefore 
g 
approximately ; 
f 20 .4 
r = a|l+g--jF(0p)j, 
the equation to any instantaneous surface of equal pressure, supposing its actual 
instantaneous and statical forms and positions to be the same. It is a prolate sphe- 
roid, of which the axis is in the line joining the centres of the earth and sun. This 
surface may be considered as fixed Avhile the earth revolves in its diurnal motion, 
during which each fluid particle (according to our approximate hypothesis of its mo- 
tion) will always remain in the same surface of equal pressure ; and if r, 6 and <p be 
taken as the coordinates of any one particle during its diurnal motion, 0 (its angular 
distance from the pole of the earth) will remain constant, and will equal the an- 
gular velocity of rotation. 
The corresponding variation in r will be obtained by 
