84 
If the expression 
be integrated we get 
r<-i 
2 V 
v = Y 0 +—ze 
V 
1 + 
2 
?} + 
2 2 
2 4 + 
2 3 
7 r 
1-3 1-3 1*3. 5 *7 
-z 6 + <fec. 
X 
where z — — — == * Taking Mr. Darwin's value for v in order 
that it may give the point of maximum cooling of the mass 
at any given time reckoned from the commencement of 
cooling (f = \ or x 1 = 2 Jet) the above becomes 
2V / 1 1 1 
v = Vo 
s/2 
7,( 
1 + l-3 + 
1'3’5 1-3-5-7 
+ &c. 
) 
2-8213722V 
" V ° + v'Tx 2-7182818 x 3 : H"l5926 
V, + V, V, - V, 
= V„ + -6826894V or putting V 0 = V— a - V = * 2 
z z 
= -8413447V! +T586553V 2 . 
where VDhj are the initial temperatures at commencement 
of cooling, a very surprising result, showing that the tem- 
perature of the point of maximum cooling of the mass is 
constant and that if Mr. Darwin’s assumption as to time is 
to be admitted, at about 100 miles below the earth’s sur- 
face there is a temperature -f-ths of what pervaded the mass 
when cooling commenced. 
Mr. Darwin’s deduced relation x* = 2 id must be taken 
with a qualification. It will give the plane on which the 
maximum cooling of the mass is taking place at any time, 
but it will not give the time at which the maximum cooling 
takes place in any plane. 
To show this take the equation which he deduces for the 
rate of cooling, viz. 
x 2 
civ _ V x c ~i Jt 
dt 2 y/ 7r/d'l 
log ( - sf ) - logC + loga; - ~ m 
