54 
PROFESSOR G. H. DARWIN ON THE MECHANICAL CONDITIONS OF 
Then we have 
o 
h 
X , w 
log - = suppose, 
where q is rigorously equal to — Sy/Ag; but for computing the approximate value 
log (w/wq) is to be employed. 
In order to proceed to the evaluation of the mean mass at various distances, we 
must assume some form for f{x). 
T assume, then, that 
/(.r) = - X). 
It is easy to show that 
•M 
f {x) dx = 
0 
1 fM _ 
and — X fix) dx — AM. 
Jo 
Hence, the mean mass = -|M, and the maximum frequency is for masses equal 
to m^^. 
Then, by (56), we have for the mean mass at radius r 
Hut 
Hi — 
I x" (]\I — .r) 
Jo ^ 
a; (jM — x) c,vdx 
■xf (M — x) e'^'-'dx = — — 4Mq + 6) — 2 (Mq + 3)1,1 
•*0 2 [ 
fM I 
x{M — x) e'i^dx = 3 (Mq — 2) + (Alcy + 2)]. 
j 0 1 
(58) 
It may be remarked that, if Mq be treated as small, we have the first of these 
integrals equal to (1 + fMq), and the second equal to ^1^^ (1 + the 
ratio of the first to the second is AM (1 + lA) M<2)- 
In order to evaluate m, we proceed to introduce the approximate value for q. 
Now, 
m 
and e% = 
then, wilting for Ijrevity, 
we have 
P= log 
2 \w, 
w 
_ 4P + 6) _ 2 (7> + 3) 
■in 
i 1 r p • / w \ MjV 
J- ( \ (P _ 2) + (P + 2) 
Wn/ 
(59) 
