784 
MR. J. LARMOR ON A DYNAMICAL THEORY OF 
and in it replace the variations of the velocities by the variations of the corresponding 
co-ordinates, we shall have just obtained this virtual work. This function ^ may 
now be expressed in terms of any generalized co-ordinates that may be most con¬ 
venient to represent the configuration of the system for the purpose in hand, and the 
virtual work of the viscous forces for any virtual displacement specified by variations 
of these co-ordinates will still be derived by this rule. “But although in an important 
class of cases the effects of viscosity are represented by the function the question 
remains open whether such a method of representation is applicable in all cases. I 
think it probable that it is so; but it is evident that we cannot expect to prove any 
general property of viscous forces in the absence of a strict definition which will 
enable us to determine with certainty what -forces are viscous and wdiat are not.”* 
89. The general variational equation of motion of the viscous system will in fact be 
|(8T - 8W- 8'.:|F) cU = 0, 
wherein 8 represents variation with respect to the co-ordinates and velocities of the 
system, while 8' represents variations with resj)ect to the velocities only, the diffe¬ 
rentials of the velocities being in the result of the latter variation replaced by diffe¬ 
rentials of the corresponding co-ordinates.t 
90. The imj)ortance of this analysis in respect to problems in the theory of radiation 
is fundamental. If a radiation maintains its period of vibration unaltered in passing 
through a viscous medium, it follows necessarily that the viscous forces of the medium 
are of the type above specified. If the elastic forces were not linear functions of the 
displacements and the viscous forces linear functions of the velocities, the period of a 
vibration would be a function of its amplitude ; and thus a strong beam of homogeneous 
light, after passing through a film of metal or other absorbing medium, wmuld come 
out as a mixture of lights of different colours. So long as we leave on one side the 
phenomena of fluorescence, we can therefore assert that the la^vs of absorption must 
be such as are derivable from a single dissipation function, of the second degree in the 
velocities, which is appropriate to the medium. 
* Lord Rayleigh, ‘ Theory of Sound,’ § 81. [An extension of the range of the function is easy after 
the method of Lagrange, loc. cit. It is worthy of notice that we can also formulate a function of mutual 
dissipation between two interacting media.] 
t It may be observed that the use of this variational equation would form the most elegant method of 
deriving the ordinary equations of motion of material dissipative systems in which the value of Jlf is 
known. For example the equations of motion of a viscous fluid in cjdindrical, polar, or any other type 
of general co-ordinates, may be derived at once from the expressions for the fundamental fnuctions in 
these co-ordinates, without the necessity of recourse to the complicated transformations sometimes 
employed. Cf. “Applications of Generalized Space Co-ordinates to Potentials and Isotropic Elasticitj',” 
‘Trans. Camb. Phil. Soc.,’ XIV., 1885. 
