336 



Sir William Thomson on the Motion of 



which, with the corresponding equation for f &c, and with (11) 

 for k, k?, &c, are the desired equations of motion. 



6. The hypothetical mode of application of K, K', ... (§ 1) 

 is impossible ; and every other (such as the influence of gravity 

 on a real liquid at different temperatures in different parts) is 

 impossible for our ideal " liquid" — that is to say, a homogeneous 

 incompressible perfect fluid. Hence we have K = 0, K/ = 0, and 

 from (11) conclude that k, k\ . . . are constants. [They are 

 sometimes called the " cyclic constants" (V. M. §§ 62-64) .] The 

 equation of motion (15) thus becomes simply 



dt d<p 



^{•(S-fMI'-fK-} 



+ &c. 



with corresponding equations for rj 

 relations from (7) between f , tj 0) . . 



dQ 



7. Let 



ffo 



-=zfa 



drjo 



g), and with the following 

 and yjr, 0, . . . 



dQ 



= *' Wo = 6 '^ 



(da d/3\ .[da! d/3'\ , 1 



\dd>~W + \M -^; + & ^bedenotedby{^t}, 



,dcf> d-^r, 

 so that we have 



Jyfr dyjr 



(17) 



(18) 



(19) 



These quantities, {</>, fa\, {0,ijr}, &c, linear functions of the 

 cyclic constants, with coefficients depending on the configuration 

 of the system, are to be generally regarded simply as given func- 

 tions of the coordinates fa <f>,0,...; and the equations of mo- 

 tion are 



W + Ta?-^ fWM6, <^+&c. = ©-5®. 



df 



cte 



(20) 



In these (being of the Hamiltonian form) Q is regarded as a 

 quadratic function of f , rj 0) t . . . with its coefficients functions 

 of yjr, </>, 0, &c. ; and applied to it indicates variations of these 

 coefficients. If now we eliminate f , ij , f , . . . from Q by the 



