NOTES ON STATICS. 549' 



15. The formulas of §13 follow immediately from those of the 

 preceding section, as does also the pi'operty proved in § 11. Thus six 

 times the volume of the tetrahedron 



-= Fi F2 {pi -{- Pi) sin i|/ ' 



^= Fi Fn {pi -\- p-i ) (sin ^1 cos 02 + cos ^3 sin 02 ) 

 i S R—Q , Q S \ 



= iPi + P2 ) {R-Q + Q)S 



= ER, 



where aS' = — — K . 

 Pi Ji 



16. The principle of Virtual Velocities. 



Lemma. The most general displacement of a body can be efiected 

 by supposing it to move along and around a certain straight line. 



Suppose that a body is at rest under the action of the forces 

 Pi, Pjj -^3} • • • • -Pn ^^ t^6 points A-^, A2, .... A^, and that the forces 

 make angles 6^^, 6^, d^, .... 6^ with the above line; then the sum o£ 

 their components parallel to this line vanishes, as does also the sum 

 of their moments around it. 



(i). Let the displacement be along the line through the distance 

 A^B^== A.B^ = ... = 8t; then 



2 (PcosS) = 0, 

 5 (P Sr cos 0) = , 

 or 2 (P 5jo) = (1) , 



where Sj^ is the projection of dr on the direction of P^, &c. 



(ii). Let the displacement be around the line through the angle da, 

 so that Pi now acts parallel to its former direction at Cj, where 

 B^ Ci = 8si ^ a^ da, if a^ denote the distance of B^ from the central 

 axis. Also let f-^ be the angle between a^ and jhj 'the distance of the 

 component Pi sin 0^ at Pi from the central axis. Then 



S {Pp sin e) = 0, 



S {Pa cos sin 6) = 0, 



— 2 (P ^s cos sin 6) — 0, 

 da 



S (P Ss cos (p sin 6) =-- 0. 

 But cos <p^ sin 01 = cos </\, if (/\ be the angle between BiCi and the 

 direction of Pi ; therefore 



2 (P Ss cog ;//) =. 0, 



or ^{PSp') = (2). 



7 



