and 



so that 



102 Lord Ray lei gh on the Law of 



is the kinetic energy, so that 



T = i w 2 + i*; 2 = ii 2 + i2/ 2 (4)* 



S may here he regarded as a function of the initial and final 

 coordinates ; and we proceed to form the expression for SS 

 in terms of hoc', By r , Bx, By. By (3) 



SS= ^(BT-BY)dt, (5) 



\BTdt = Ux\Bx + yBy)dt 



K. dBx . dSy\ 



= \ x Bx + y By\ — \ (x Bx + y By) dt; 



£S = ["* Bx + y By] '- f \x Bx + ySy + 5 V) dt. 



By the general equation of dynamics the term under the 

 integral sign vanishes throughout, and thus finally 



B8 = uBx + vBy-u'Bx'-v , By / (6) 



In the general theory the corresponding equation is 



BS^pBq-Sp'Bq' (7) 



Equation (6) is equivalent to 



u' '== — dS/dx', u = d$/dx, ~1 /gx 



v' = ~d$/dy', v=dS/dy. J 



It is important to appreciate clearly the meaning of these 

 equations. S is in general a function of x, y, x', y' \ and 

 (e. g.) the second equation signifies that u is equal to the 

 rate at which S varies with x, wften y, x' ', y' are kept const ant , 

 and so in the other cases. 



We have now to consider, not merely a single particle, 

 but an immense number of similar particles, moving inde- 

 pendently of one another under the same law (Y), and distri- 

 buted at time t over all possible phases (x, y, u, v). The 



* As is not inmsual in the integral calculus, we employ the same 

 symbols x } &c. to denote the current and the final values of the variables. 

 If desired, the final values may be temporarily distinguished as x" , &c. 



