Equilibrium and Initial and Steady Motions. 219 



x' f y', z' denote any other velocities consistent with the con- 

 nexions of the system, the Principle of Virtual Velocities gives 



Sif-w)/+PJ-^)j r +(K-to)i'[ =0, . (1) 



by means of which the initial velocities k &c. are completely 

 determined. 



In equation (1) the hypothetical velocities at &c. are any 

 whatever consistent with the constitution of the system; but if 

 they are limited to be such as the system could acquire under 

 the operation of the given impulses with the assistance of mere 

 constraints, we have 



2T' = 2m(i' 2 + y' 2 + *' 2 ) = 2 (P*' + Qy' + R*') .. . (2) 



This includes the case of the actual motion. 



Returning now to the general case, suppose that E' denotes 

 the function 



E'^SlPi'-hQ^ + RiO-iSm^ + ^ + i' 2 ), . (3) 



becoming for the actual motion 



E = T=i2H;. 2 + y 2 + i 2 ), • • • • (4) 

 or, for any motion of the kind considered in (2), 



E' = T'=i2™(^' 2 + y' 2 + i' 2 ). ... (5) 

 For the difference between E' and E in (3) and (4), we get 

 E -E'= \%m{^ + y 2 + i 2 ) + i^mft'* +y' 2 + i /2 ) 

 -^(P^ + Qy' + Ri'), 

 in which by (1), 



2 (Pi' + Qy' + Rz 1 ) = &■(« i + yy' + £ i') ; 

 so that 



E-W=i2m\(i-i'f+(y-y')z+(z-z')^ f . . (6) 



which shows that E is a maximum for the actual motion (in 

 which case it is equal to T), and exceeds any other value E' by 

 the energy of the difference of the real and hypothetical motions. 

 From this we obtain Bertrand's theorem, if we introduce the 

 further limitation that the hypothetical motion is such as the 

 system can be guided to take by mere constraints ; for then by (5) 



E-E'=T-T'. 



By means of the general theorem (6) we may prove that the 

 energy due to given impulses is increased by any diminution 

 (however local) in the inertia of the system. For whatever the 

 motion acquired by the altered system may be, the value of E 



