MR. HENNESSY’S RESEARCHES IN TERRESTRIAL PHYSICS. 
545 
16 f 1 
PQ' ~~P'9 
15\q' — <^q ^ 'Ijxr—p 
q’-qcr 
pq'-qp’ P’^-P' ^ 
2p<T—p\\\ Sq'—qa-J {<l' — °'q)^ 
\ 3 7 ^ 
putting JO =5 [(e) — e], p'=3[(e) — e], q'=de, q=5{2e—m). 
2 2)(j — 
But 1 — g q—^ always less than 1, and q —(^g, pg'—p'g af<i both positive and 
small quantities. Hence is positive, and therefore the value of a, deduced by 
dd I 223^ 
making -^*= 0 , gives a minimum for a,. This value is evidently a— 
On substituting this in the value of a^, we shall have ai = 0 : ^‘=0 would be also 
satisfied by making < 7 = 00 , but the supposition that v is finite and little different from 
1 , excludes this value. Hence must increase from its least value as a decreases to 
its least value, but this is when < 7 = 1 , or when the shell is supposed to be homo- 
geneous ; consequently the greatest value of will be determined from the equation 
K-\--m — 2e 
4 
3(|».-c) 
_2 , 1 i”-*') 
~3 '35 
- m — e 
4 
From Section IX. (e)=^ nearly; and using for e and m the values respectively 
and we shall obtain 
<=i[2+S3=-®77.4; 
hence ai='99539, 1— a,='00461, and therefore, consistently with observation, the 
least thickness of the earth’s crust cannot be less than 18 miles. 
From Section III. it appears that as {k) diminishes the more will the constitution of 
the nucleus and shell be different, and consequently the least value which can be 
attributed to {k) from observation, may be assumed to give the least value of in 
equation (35.). By using this equation together with (29.), I find, after a few trials, 
that with ai=*85, the approximate value of Q would be *023441. But Q increases 
with a„ hence the greatest thickness of the shell cannot, on the above assumption, be 
greater than ‘IS, or 600 miles nearly. 
(3.) The EartKs Ellipticity when entirely fluid. 
25. By the aid of equation (11.), Part I., the following values of the earth’s primi- 
tive ellipticity have been obtained with different values of {k). 
{k)= -7481, E: 
{k)= -896, 
(k)=V, 
E= 
E= 
302 - 22 ' 
1 
310 - 74 ’ 
1 
316' 
With 
