HAroHTON — On Complete Tidal Equaiions. 235 



In addition to the foregoing differential equations of motion, the 

 geometrical equation called the "Equation of Continuity" has to be 

 found. 



I here give it by means of an investigation more general and more 

 simple than that used by Laplace and Airy. 



Let 



w = the linear velocity in the direction of the radius vector ; 



V = the angular velocity in latitude in the plane of the moving 

 meridian ; 



w = the angular velocity in longitude. 



If we imagine a prism erected on a trapezoidal base whose four 

 comers are 



(!)• e',^'-, (2). e' + de',<i>'; 



(3). 6', <j>' + dcf>' ; (4). 6' + dO', ^' + dcj>'', 



the sides (1, 2) and (3, 4) are equal, and each rd$' ; but the sides 

 (1, 3) and (2, 4) are not equal; the first being r sin 6'«?<jf)', and the 

 second being r sin {6' + dd') dcfi', or 



r (sin 6' + cos 6W) dcf>'. 



If, now, z denote the variable depth of the sea, the quantity of 

 water passing in the time dt, through the wall of the prism (1, 2), will 

 be 



z X rd6' X r sin 6'wdt. 



The quantity of water passing in the same time through the wall 

 (3, 4) wiU be 



z+ — d4>'\x rdO' X r sin 6' iw + —, dcfi' \dt. 

 The difference of these quantities is 



r^ sin 6' dO' dt [w^ + z ^X^', 

 \ dtp d(p J 



or, 



r' sin 6' dO' dcj>' dt^^. (a) 



d(p 



The quantity of water flowing in the time dt through the wall 

 (1, 3) is 



s X r sin 6' dcji' x rv dt, 



