of Perfectly Elastic Spheres. 25 
to prove that the mean vis viva of each particle will become the 
same in the two systems. . 
Let P be the mass of each particle of the first system, Q that 
of each particle of the second. Let p, g be the mean velocities 
in the two systems before impact, and let 
p',q be the mean velocitiesafter one impact. 
Let AO=p and O B=q, and let AO B be 
aright angle ; then, by Prop. V., A B will be 
the mean relative velocity, O G will be the 
mean velocity of centre of gravity; and 
drawing a G é at right angles to OG, and 
making aG=A G and 6G=BG, then Oa 
will be the mean velocity of P after impact, 
compounded of OG and Ga, and 0d will 
be that of Q after impact. 
Now 
a ee 2 ———— 
AB= vVp?+¢?, AG= pa Vp?+q?, BG= faa Vp?+@q, 
oga VPP +Q'¢* 
P+Q 
therefore 
paOca ee ne ees 
P+Q 
and 
Fe We ee ee 
P+Q 
and 
Py?—Qg"= (Foe) (Pe Qe) - pcos iiteae(D) 
It appears therefore that the quantity Pp?—Qg? is diminished 
at every impact in the same ratio, so that after many impacts it 
will vanish, and then 
Pa? =Q¢?. 
ot lei 3m 3m 
Now the mean vis viva is 3 Pat @ Pe for P, and 3 Qgq? for 
Q; and it is manifest that these quantities will be equal when 
Pp? =Q¢’. 
If any number of different kinds of particles, having masses 
P,Q,R, and velocities p,q,7 respectively, move in the same 
vessel, then after many impacts 
Pp?=Qg?=Rr?, &. . . . . . (7) 
Prop. VII. A particle moves with velocity r relatively to a 
number of particles of which there are N in unit of volume; to 
