206 
MR, Gr. H. DARWIN ON THE STRESSES 
must now consider the factor by which it is to be multiplied, in introducing the height 
of the mountains and gravity. 
This factor is computed in § 4; it is there shown that if r=a-\-h<Si be a harmonic 
spheroid, the factor is — 2({ — 1) gwh/ (2i +1) ci ! . 
Now if the harmonic i be of an infinitely high order, 9 ; becomes simply cos z/b, and 
the equation to the surface is 
—h cos 7 
b b 
£ being measured downwards. Thus the harmonic spheroid As,- now represents a 
series of parallel harmonic mountains and valleys of height and depth h , and wave¬ 
length 27 rb. 
The factor becomes —gwh/a\ when i is infinite. 
Thus the effective disturbing potential W, which is competent to produce the same 
state of stress and strain as the weight of the mountains and valleys, is given by 
W— —givhe~^ b cos ^ . ..(36) 
Now revert to the expressions (ll) for the stresses. 
When i is infinite J= 2 i 3 , and they become, on changing x into (a—£) 
P= 
T= 
3 
2i- ' + 2 d% + '2i"’ 
1 It e\ dW 
s (“-£)- 
--(a 2 -r») 
d?W 
3 ( 
\l\dz 
2 i 2 \ 
dz 
Now as shown above a 3 —r 3 = 2 a£ and a/i—b in the limit; making these substitu¬ 
tions, and dropping the terms which become infinitely small when i is infinite, we 
have 
and by a similar process 
c n w d? W 
•p n „ 
E =^> Q=0 
Then from (36) and (37) we have 
P= —givh^e~ i/h cos | 
R= gwh~€~t'' h cos Z - y .(38) 
T= givh^e~ m sin ^ 
Since the stress-difference 
A = y(P-R) 2 +4T 3 
we have 
4=2 gwhje~* /b .. . , . . (39) 
