D. Trowbridge on the Nebular Hypothesis. 355 
in which y is the distance of any particle from the axis of rota- 
tion, From (13) and (14) we have 
creases with the loss of heat, we see that as the solar spheroid 
gradually cools down, the living force of the mass gradually in- 
a eases. Or in other words, as the body loses its heat, the at- 
F traction of gravitation has a greater influence and increases the 
. __ living force of the body. 
: ven if the angular velocity of all the parts is not the same, 
we may arrive at an approximate result, provided the different 
strata differ but little in angular velocity from the mean. Let 
“ represent the mean angular velocity of the body, then the 
true angular velocity may be represented by w-+-w’, wit w, &e., 
in which we shall suppose w’, w”, #”, &c., so small that we may - 
neglect their squares in comparison with »?. We hence have 
2 (mv?) = my? (ato )2 =F (ma? +2mow') y?=w? Mk? +2u5(mo'y?)=W 
and 2m(w--w')y? = aMk? + 2(moi'y?) — A, 
Therefore 
w= 2Aw—w?Mk? = of 2A —oMk?], foe ew (EOP 
From this equation we also see that W increases with. We 
May also have 
W = Ao-+w (ma'y?). 
_ Either of these expressions for W will approximate to Aw as 
_, &e., diminish without limit. From this we conclude that as 
S00n as a rotation has commenced in a nebulous mass, its living 
force will increase as the mass cools. 
much width; but the materials being somewhat heterogeneously 
* If different strata have different angular velocities, A will no longer be a con- 
- Jo find an expression for the angular velocity in this case, of any one stra- 
and 
tum, let W unctions of r, such that dr is the thickness e stratum 
= angular isw. The vis viva of the part immediately within the 
Stratum of thickness dr will be W— r, and the su the areas will be 
—D; Aér. We hence have W—Dr Wér=wA—oDr Adr; or by (15) Dr W= 
Dr A,a=P*Y —_p,w, This result js obtained by supposing the spheroid 
ca C Dr A : is oi 
pant the stratum $r to have the same pee all its parts; and that 
y is the same as that of the s 
