702 Dr. C. V. Burton on the Kinetic 



of inertia, and it is to be understood that o> 1} a> 2 , eo 3 are to be 

 replaced by their values in terms of the time-fluxes of the 

 angular coordinates of the system. Of the coefficients, 

 M and I : alone are constants, I 2 , I 3 , I , and IF being functions 

 of 6. In the absence of any external forces acting on the 

 system, T is the Lagrangian function, which we proceed to 

 modify with respect to the coordinates %, %'. The angular 

 momenta of the governors about their axis are 



and BT/^=^+o )l ) = 0(sa y ), V 



Thus 



C C 



x^j-"** x=j-<»i ( 24 > 



The modified Lagrangian function is thus 



T^T-Cx-CY 



=iM(P + ^+P) + il^ + JI^ + JL^ + ilo^ 



+ 2 it + 2 if 



c (r--)- c '(?"4 



i. e. 

 V = l M(P +f+ ?) + ^ W + 4 w + i I„,« + i I> 



28. Since the rotatable governors are acted upon by no 

 forces having a moment about the axes A A or A'A', and 

 since the coordinates %, %' do not appear in the coefficients 

 of the expression T, the whole kinetic energy may be divided 

 into two parts : one (to be .called 'C) being a h.q.f. of the 

 velocities x, y, z, w u w 2 , &>3, #, and the other (to be called K) 

 being a function of the coordinate 0, and involving besides 

 only the constant momenta C, C. (25) may in fact be 

 written 



T' = C-K + (C + C> i; .... (26) 



and we shall accordingly be able to treat the rotational 

 energy K of the governors as potential, provided the constant 

 angular momenta C, O are equal and opposite. 



29. Alternatively, instead of the two governors being ro- 

 tatable independently of one another, we may suppose them to 

 be positively connected in such a way that their rotations with 

 respect to the frame B are always equal and opposite. The 



