66 Demonstration of the Principle of Virtual Velocities. 



tion, even when conducted with the greatest possible care, is 

 destructive, hyponitrons ether and gaseous matters being the 

 principal products obtained. Nor were we more successful in our 

 attempts to procure a sulphurous or hyposulphuric ether by the 

 same process. 



Art. V. — A new Demonstration of the Principle of Virtual Ve- 

 locities ; by Prof. Theodore StronS, 



Let any body or system of bodies, (or material points,) be af- 

 fected by the forces P, Q,, R, and so on j imagine points consid- 

 ered as fixed to be taken in the lines in which 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, q the distance of the 

 point of application of d from the point taken 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 j», 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 dis- 

 tances j9, q, &c. 



We shall denote the sum of the products Pja, Q,q, &c. by M, 

 and we shall have Pj? + Q,g4-&c.=M, (1,) then if the forces bal- 

 ance each other, p, q, &c. will each be constant. 



We shall suppose that p, q^ &c. are each constant, and that P, 



Gl, &c. become P + P', Gl + Q,', &c. but that p, q, &c. are yet 



each constant ; also that M becomes M-f M^ ; then (1) will become 



(P+F)p+(a-t-Q<0?4-&c.=M+M', which by (1) reduces to 



M' 

 P'i?+Gl'9'+&c.=M', (2); if we multiply (1) by j^j, we get 



PM' aM' 



i?-j^- +9^-1^1- +&c.=M', which must evidently be identical with 



(2), so as to leave jo, q., &c. each arbitrary, hence the coefficients 

 of p must be equal, also those of q must be equal, and so on : 



P' M' Ql' W 

 *'*F~M' Q~M' ^"^ ^° °"- Hence it is evident that P', 

 Q,'j &c. have all the same sign, and that they have the same pro- 



