504 VISCOSITY. [CHAP. XI 



and &a = -&r = i (# -#) .................. (C), 



the typical equation (1) takes the form 

 d dT dF 



provided 



2F=b u qi + b 2 ,q/+... +26 la g,g a + ............... (8). 



From the equations in this form we derive 



%- t (T+V) + 2F=2Q r q r .................... (9). 



The right-hand side expresses the rate at which the extraneous 

 forces are doing work. Part of this work goes to increase the total 

 energy T-f V of the system; the remainder is dissipated, at the 

 rate %F. In the application to natural problems the function F is 

 essentially positive : it is called by Lord Rayleigh *, by whom it 

 was first formally employed, the Dissipation-Function. 



The terms in (7) which are due to F may be distinguished as 

 the frictional terms. The remaining terms in q ly (J 2 , ..., with 

 coefficients subject to the relation /3 rg = ft fr , are of the type we 

 have already met with in the general equations of a gyrostatic 

 system (Art. 139); they may therefore be referred to as the 

 gyrostatic terms. 



279. When the gyrostatic terms are absent, the equation (7) 

 reduces to 



d dT dF dV 



-T. -j-r- + -T^ +-F- =Q, .................... (10). 



dt dq r dq, dq r 



As in Art. 165, we may suppose that by transformation of 

 coordinates the expressions for T and V are reduced to sums of 

 squares, thus : 



2r = o 1 g 1 2 + a 2 g a a + ..................... (11), 



27=^+0^ + ..................... (12). 



It frequently, but not necessarily, happens that the same 

 transformation also reduces F to this form, say 



2F =1^ + 14?+ ..................... (13). 



* &quot; Some General Theorems relating to Vibrations,&quot; Proc. Lond. Math. Soc., 

 t. iv., p. 363 (1873) ; Theory of Sound, Art. 81. 



