194 Dr. C. Y. Burton on a Theory 





/ 



9 



h 



OL 



h 



m 1 



n x 



fi 



k 



m 2 



n 2 



7 



k 



m 3 



n 3 



(3) 



Let the original density of the aether be taken as unity ; and con- 

 sider that portion of sether which, in the undisturbed condition 

 of the medium, would have filled the vol lime- element d%dr)d% 

 with coordinates (£, rj, f). Since/, g,h are the displacements 

 parallel to Ox, Oy, Oz, of the 'portion of cether so defined, f g, h 

 will be the actual velocity-components of this portion, and the 

 energy due to the motion of the strain-figure is 



T=i^(r+f + h')^dnd^. .... (4) 



Now let the axes flf , £lrj, X2f be instantaneously coincident 

 in direction with Ox, Oy, Oz respectively, so that 



l 1 — m 2 = n 3 =l; l 2 = l 3 = m 3 =m 1 = n l = n 2 = 0, . . (5) 

 and 



Z 1 = ??i 2 = n 3 = ; n 2 =— m 3 =fi> 1 ; l 3 =— w 1 = &) 2 ; m 1 =—l 2 = a 3 , (6) 

 where co^ co 2 , &> 3 are the angular velocity-components of the 

 strain-figure about the axes of £, rj, £ or of x, y, z ; relations 

 which, in general, are only instantaneously true. 

 From (3), 



/= \ti + 1 2 $ + Z 3 y + I x ol + 4/3 + Iff; 

 and writing 



5i =ai; §7r as; ^* s;&c -' * * (7) 



the last equation becomes 



/= h (*i? + «2^ + « 3 ?) + h (ftf + fti + &£) + / 3 (7i £ + 72^ + y 3 ?) 



+ /i«+/ 2 /9 + Z 3 7. ... (8) 

 Again, from (1), 



f*=M*"X)+fii 1 (y-Y) + n 1 (^Z), 



so that 



f = ^i(^if + ^ + ^3?) + ™i (™i? + "V7 + »i 3 ?) + %(>i£ + ?? 2 7? + n£) 



-^X-miY-^Z; 

 or, using (5) and (6), 



£=<w 3 ?7 — w 2 ? — X. 

 Similarly 4«..S-«tf-t, 



and • • 



C=w 2 f— a)!?; — Z. 



