69G Dr. C. V. Burton on the Kinetic 



11. Following Kouth *, let us now modify the function T 

 with respect to the coordinates %, x\ • • •> the modified function 

 will be 



~dx x B% /% " " * ' [) 



the %'s being supposed eliminated from the right-hand of (7) 

 by means of the equations 



BT/3x=C, 3T/dx'=C', (8) 



so that the C's are the momenta corresponding to the j^s 

 respectively, 



12. In a system such as we contemplate, the energy is 

 made up of two parts : one a h.q.f. (C) of the velocities 

 i/r, $, #, . . ., and the other (K) independent of those velo- 

 cities ; that is 



T = C + K (S) 



In this case 'C is the energy which we recognize as kinetic, 

 and K is that which we call potential. 



1 3. It is of course understood, when yjr, <p, 0,... are the 

 only " working coordinates/' that the system is acted upon 

 by no external forces except *&, O, ®, . . . ; in these circum- 

 stances it is shown in Kelvin and Tait that (9) will be fulfilled 

 provided the %'s do not appear in the coefficients of the 

 energy expression T ; and it is easy to show that (9) will not 

 be fulfilled otherwise. Hence in place of (9) we may write 

 the conditions 



3T/d % = 0, BT/B % ' = 0, (9a) 



14. Kelvin and Tait's analysis f relating to the ignoration 

 of coordinates is therefore applicable, and we have 



C, C, . . . ail constant ; . . . . (10) 



while K is a h.q.f. o£ the C's. 



15. From the modified function T' the equations of motion 

 of the system may be obtained in the form 



^ g>T / _BT / 



dtty W ' (11) 



with similar equations for <3>, ©, . . . ; and these by a process 



* ' Rigid Dynamics/ vol. i. chap. viii. 

 t Loc. cit. 



