116 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



d (du\ _ d /dm\ ._. 



J~z \d~t) ~Tx \dt) 



From equation (2) we get 



d (du . du du du\ d (dv dv dv dv\ 



-s-[-ji+ *?- 9-?- +#-5-] = -j-[ -r.+ ' u -r+ v 5- + m -r ) 



dy\dt dx dy dz) dx\dt dx dy dz) 



d 2 u d 2 u du du dv du d 2 u dw du 

 dtdy dxdy dx' dy dy'dy dy 2 dy dz 



d 2 u d 2 v d 2 v du dv dv dv d 2 v dm dv 

 + m- =- - + ___ +__.__ + __. + - _+ 



dydz dxdt dx 2 dx' dx dx' dy dxdy dx dz dxds 



But ( ) .-7^=. j + u , j + v ^r-^ + w ^r 



\dt/ dy dydt dxdy dy* dydz 

 fd\ dv d 2 v d 2 v d 2 v d 2 v 



. - I l*t _ , _L A* __ .A. J _ 



\dt) ' dx~dxdt dx 2 dxdy dxdz 



(d\ du du du dv du dm du 

 dt) dy dx' dy dy ' dy dy ' dz 



(d\ dv du dv dv dv dm dv 

 dt) dx dx ' dx dx dy dx ' dz 



or from equation (1), 



(d\ du dm du dm du 

 dt) ' dy dz' dy dy ' dz 



(d\ dv dm dv dm dv 

 dt) dx dz dx dx' dz 



(d\ /du dv\ _dn /du dv\ dw dv _dm_ du 

 d~t) \dy~dx) ~~dz \d~y~ ~dx) ~dx ' Jz~~dy ' Tz 



(d\ fdu dm\ _dv /du dm\ dv dm dv du 

 dt) \dz dx) dy\dz dx) dx' dy dz' dy 



(d\ /dv dm\_du/dv dm\ du dm du rf 

 dt) \dz dy) ~ dx\dz dy) dy ' dx dz ' dx' 



Tf _5? -- ]yj 



dz dy 



dm du_-^ 

 dx dz~ 



du dv 



- -j-=P we get 



dy dx 



(dP\ _ p dm dm dv dm du 

 dt) dz dx ' dz dy dz 



(?N\ T^dv dv du dv dm 

 i N u . 

 dt) ' dy dz' dy dx ' dy 



