426 
PROFESSOR G. H. DARWIN ON FIGURES OF 
If, however, k = 2, there is on the left-hand side an additional term 
3a) 2 „ faV , /A A 7 4! _ »* / e\ 3 6 _ , /c\ 3 
16*- \c/ ' 2 \ c / 1210! 877 - ^ y 4.3 15 hi/' 
The equation of dV/dPi to zero gives a similar equation. 
Now, if we put hk + Ik for ni, except when k — 2, and then put h 2 + l 2 + (c/A ) 3 = n 2 , an( i 
similarly introduce the H’s and A’s ; and if we put fk — mu, except when k = 2 , and then put 
p 2 = m 2 + T \ e (c/A) 3 , and similarly introduce the ATs, it is easy to see that the equations (i.) to the 
two surfaces become the same as (72), and the equations of condition between n and N, and between 
p and P, become exactly those which we found by a different method above in (23), (44), and (59). 
The only difference is that the equations for h and l are fused together. 
This, therefore, forms a valuable confirmation of the correctness of the long analysis employed for the 
determination of the forms of equilibrium. 
The formula (viii.) also enables us to obtain the intrinsic energy of the system, that is to say, the 
exhaustion of energy of the concentration of the matter from a state of infinite dispersion to its actual 
shape, with its sign changed. 
The last line of (viii.) depends on the yielding of the fluid to centrifugal force, and must be omitted 
from the exhaustion of energy. 
Besides the rest of (viii.), we have in the exhaustion of energy of the system, the exhaustion of the 
two spheres and their mutual exhaustion. 
It is clear, then, that the intrinsic energy is 
-cs^mc^+^-G’O 8 — 
where the n, N, P, p, have their values determined in accordance with the condition that the surfaces are 
level surfaces. 
In evaluating the intrinsic energy from this formula, it is convenient to refer the energy to that of a 
sphere of such radius, b, that its mass is equal to the whole mass of the system. Then 5 3 = a 3 + A 3 , 
and we may take the intrinsic energy as 
a-) 2 v (* - 1). 
Thus i will be a numerical quantity which is positive. 
I find from (ix.), with (3 = and a = A, that i = '4873. 
With regard to the kinetic energy of the system, we have seen in § 10 that the moment of momentum 
is (|tt)? 5 5 X /<, where /i is a numerical quantity, and w r e find in the course of the determination the 
function 3 a> 2 / 47 r. Then, since the energy is half the moment of momentum multiplied by the angular 
velocity, it is clear that the kinetic energy is 
The kinetic energy, as represented by e = (3«> 2 /4-)’-, is comparable w r itli the intrinsic energy as 
represented by i. 
In the case of ji — and a = A, I find the kinetic energy e = '0925. 
Thus the total energy E = e + i — '5798.* 
* [See foot-note added to the Summary above. October 10, 1887.] 
