G. F. Frrzanratp—On the Mechanical Theory of Crookes’s Force. 59 
Proceeding similarly we can get the tangential pressures on these areas, and we 
easily see that they are 
P,,.=P,=Mnv*(1 - p?) sin ¢ cos 
Per Pre=MSnv2pV 1 — p?. cos 
Py=Pyp=M Env pV 1—p?. sin 6 
If now we proceed to calculate the energy carried across these areas per unit time, 
we get kn v* # as that carried across the 1st area by molecules moving in the direc- 
tion #, ¢, when k is the coefficient by which the energy of translation must be multi- 
plied, in order to obtain the total energy, calling the quantities of energy Q, Q, and 
Q., we thus get 
Q:= Mk un Q,=Mkine V1= 2? sin Q-=MeEnv? V1 =? COs 
In order to be able to perform these summations, it is necessary to know the 
mean values of nv, nv, and nv? in terms of # and ¢, and I shall, in the first place 
merely assume that they can be expanded in a series of spherical harmonies, thus : 
— No 
a (IN a IN ete I ae ee ) dude 
—. Ne 
n= (IB ae dBh db 18th 6 G50 6 6 ) dude 
ING 
aoe GoeGeGsscces ) dudes 
Tv 
The effect of this is to obtain our former results under the following simplified 
forms. Our first series of equations gives A,—o and as A, must be of the form 
A,=ap-+a, Wl—p? sin ¢ +a, /1—p? cos 
We Ch A=G=&=o 
The second system of equations gives 
Pag [BAB edudy 
P= MNO p(B. +B,)(1 = p?) sin 2g dpdg 
Pe ff(B,-+-B,)(1 — p2) 008 %9 dudp 
Bae oe ff B, (1 = p2) sin @ cos ¢ dude 
PP ip B, wV¥1— p2 cos dude 
MNv,? ———> 6 
PL =P = Agr Sf Bs pv 1 —p? sin ¢ dudp 
If now we assume 
By= b,(u?—4)-+8, (1 — p?) cos 26 + 6, (1 — p2) sin cosip 
+b, pW 1 — pw? cos @ -+ 6, pV 1-2? sin 
D0 
