504 VISCOSITY. [CHAP, xi 



and & = -&r = i (-*) .................. (6), 



the typical equation (1) takes the form 



i++A.*+***.. + ....... m. 



provided 



...+2b 1 ,q 1 q,+ ............... (8). 



From the equations in this form we derive 



(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 + V of the system; the remainder is dissipated, at the 

 rate 2F. 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 l} q. 2) ..., with 

 coefficients subject to the relation /3 rg = /3g rj 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_ n 



dtdjr + djr + dqr-^" 



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 : 



2^ = ^ + 04/4- ..................... (11), 



2V=c 1 q l *+c&+ ..................... (12). 



It frequently, but not necessarily, happens that the same 

 transformation also reduces F to this form, say 



2F=b 1 q 1 * + b& + ..................... (13). 



* " Some General Theorems relating to Vibrations," Proc. Lond. Math. Soc., 

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



