66 Demonstration of the Principle of Virtual Velocities. 
tion, even when conducted with the greatest possible care, is 
destructive, hyponitrous ether and gaseous matters being the 
principal products obtained. Nor were we more successful in our 
attempts to procure a sulphurous or hyposulphurie ether by the 
same process. — 
Arr. V.—A new Demonstration of the Principle of Virtual Ve- 
eo bined by’ eed on Stone. 
Ler any body or system of bodies, (or material points,) be af- 
fected by the forces P, Q, R, and so on; imagine points consid: 
ered as fixed to be taken in the lines in whieh the forces act, and 
let p denote the distance of the point of application of P, from 
the point taken in the line of its action, ¢ the distance of the 
point of application of Q from the point Niten ‘i in the line of its 
direction, and so on; and suppose the points to be so assumed 
that P, Q, &c. shall tend at the same time to increase each of the 
distances p, q, &c. or to decrease them, (the positions of the fixed 
points in other respects being supposed arbitrary): we shall re- 
gard each force and distance as positive, and it is manifest that 
the equilibrium consists in the relation of the forces to each other 
being such that their actions shall not alter any one of the cue 
‘tances p, ¢, &e. 
We shall denote the sum of the products Pp, Qq, &c. by M, 
and we shall have Pp +Qq+&c.=M, (1,) then if the rte bal 
ance each other, p, q, &c. will each be constant. 
We shall suppose that p, g, &c. are each constant, and ait r, 
a ‘&ec. become P+ P’, Q@+Q’, &c. but that p, ¢, &c. are yet 
each constant ; also that M becomes M+M’/; then (1) will become 
(P+P’)p+(Q+Q’)q+&c.=M+M’, which by (1) reduces to 
M’ 
ie ealillalpioee (2); if we multiply (1) by M> Wwe get 
PM QM 
Pw toy +é&e. =M’, which must exidanily. be identical swith 
(2), so as to leave p, q, &c. each arbitrary, hence the coefficients 
Tea —_ also those atepmiberrtanegad; and so on: 
-p=M’ @=M and so on. Hence it is evident that P’, 
Q’, &c. have all the same sign, and that they have the same pro- 
