Equilibrium of Vis Viva. 159 



between the exits from the two parallelopipeds is also exactly 

 ht, it follows that the condition-point must enter both paral- 

 lelopipeds the same number of times and must remain for the 

 same time in each. This gives dt' = dt, or 



f{x', y>, 6') <U dy> d6'=f{x, y, 0) dx dy dO. 

 Bat we have 





~bx f 3#' ~bx' 



3*' 3y' 30 





dx'dy r de f = 



3/ 3;/ 3./ 

 3*' dy' ^ 



W, M', M' 

 ~dx 3y 3# 



. dx dy d6 



As ht is constant, and g 2 = const. - 

 (2) give 



3^' a U "by' 

 -^— = 1 + f cos 6 . —, ^-=. 

 ov q dy 



-2V, equ 



L+9?sin ^ 



& 



30' 



3^ 



^-77 = 1 — (£ cos # + ?? sin 0) — 

 3# v 



If we neglect the terms of order & 2 , the functional deter- 



minant reduces to 



and consequently 



3*' 

 3^ 



3/ 



3^ 

 3# 



= 1, 



dx f dy f d6 r =dxdyd6, 



from which also /(&•', ?/, #') =f(x, y, 6). As we can pass from 

 the parallelopiped dxdydd to its corresponding one, and from 

 this latter to its corresponding one, and so on, it follows that 

 f(x, y, 6) is always constant, and therefore that dt = Cdx dy d6. 

 This result agrees perfectly with that found in my paper pre- 

 viously quoted, on the " Solution of a Mechanical Problem." 



Lord Kelvin denotes by N d6 dr the number of times during 

 the interval T that the moving particle traverses an element 

 of surface, whose length in the direction of motion is ds and 

 whose breadth perpendicular to this direction is dr, in such a 

 manner that 6 lies between the values 6 and 6-\-dd. As the 

 moving point always remains for a time ds/q in the element 

 of surface, the fraction of T during which the moving point 

 is situated in drds and has at the same time a direction of 

 motion between 6 and 6 + dO, is 



N 

 -drdsde. 



