330 Prof. G. H. Darwin. On Jacobis Figure of [Nov. 25, 



The function (26) is the quantity which will be tabulated. 



Thus in the tables the unit of moment of momentum is taken as 

 (|-7ro-)'a 5 , or mfa?, and the unit of energy as (-f 7rc-) 2 a 5 or ra 2 /a. 



It remains to evaluate the intrinsic energy, or the energy required 

 to expand the ellipsoid against its own gravitation, into a condition 

 of infinite dispersion. 



If dt be an element of volume, then this energy is 



IJ 



Yadt 



integrated throughout the ellipsoid. 



This will be denoted by (f 77-<r) 2 a 5 (i— 1), or m 2 a _1 (i — 1), so that i 

 will be positive. 



Now Y=Lx* + Mif + Nz* + P, and if we denote by A, B, C, the prin- 

 cipal moments of inertia of the ellipsoid, we have 



fll 



x*a dt=§(B + C - A) ={mfl 2 , 



dt=\mc*. 



and similarly, j" jj?/ 3 ^ dt—±mb 2 , jj 



Also j*j*Jo-^=ra. 



Hence —(ji-l) = T \m [La* + Mb* + No* + 5P] 

 a 



= T Vma 2 (sec /5 sec 7 )t[_L + M cos 2 /3 + N cos 2 7 + 5P<T 3 ] . 

 But if we take the values of L, M, N given in (15), and note that 



2 



P = 7ra 3 cos § cos 7 ; — F, 



a sm 7 



it easily follows that 



L+M cos 2 /3 + N cos 2 7 + Pa" 2 =0. 



Hence — (i— 1) = — f ?rea 2 (sec fi sec 7)* . Pa" 2 

 a 



= — 4??za 2 (secy3sec7)^ . }— \ • — 1 ^ 



V ^ , - / 4 a- 2 asm 7 



„ _ ± rn^ (cos (3 cos 7)^ 

 5 a sin 7 



Therefore i = l-^l^^F (27) 



sin 7 



For a sphere 7 becomes infinitely small, and F becomes equal to 7, 

 so that F/sin7 = l. Thus i— 1=— f. Therefore the exhaustion of 



