ON PHYSICAL LINES OF FOKCE. 



481 



x 



y 



d 

 j- 



d 



Sz' 

 --r 



dz' 



(64), 



with similar equations for quantities involving Sy and 8z. 



Let a, ft, y be the values of a, ft, y referred to the axes x, y', z ; then 



(65). 



y = l,a + mj$ 4- n 3 y J 

 We shall then have 8a = l l 8a+l$ft' + l 3 8y' + y0 t -ft0 3 (66), 



-ft0, (67). 



By substituting the values of a, f?, y', and comparing with equations (64), we 

 find 



(68) 



as the variation of a due to the change of form and position of the element. 

 The variations of ft and y have similar expressions. 



PROP. XI. To find the electromotive forces in a moving body. 



The variation of the velocity of the vortices in a moving element is due to 

 two causes the action of the electromotive forces, and the change of form and 

 position of the element. The whole variation of a is therefore 



8 * ............... < 69) - 



But since a is a function of x, y, z and t, the variation of a may be also written 



Sa^Sx + ^Sy+^Sz + ^fc ..................... (70). 



dx ay dz at 



Equating the two values of Sa and dividing by S^, and remembering that in the 

 motion of an incompressible medium 



a. ,_ _ n 



dx dt "*" dy dt + dz dt ~ 



VOL. I. 



61 



