Prof. Challis on a Mathematical Theory of Tides. 27 



7T 



Let -57 be the atmospheric pressure. Then, since \= +^ at 

 the points of the surface for which r=a, we have 

 ~ ma 2 , ,, N 



which equation determines the arbitrary quantity i/r (t) . It will 

 hence be found that 



+ l^s(r*-b*+^(r-b) 2 (r + 2b)y s*\cos2(d-v,t). 



By supposing that p = vr, the equation of the exterior surface of 

 the ocean at any time t is obtained. 



From this first approximation the main features of ocean-tides 

 are readily deducible. But before drawing any inferences, a re- 

 mark of considerable importance must be made, in order to meet 

 an objection that might be raised against the previous reasoning. 

 It appears that the equation (/3), viz. 



&*"*" df + ' dz* > 



is not satisfied when the foregoing value of <ft is substituted 

 in it. I have already intimated that this was to be expected, 

 because, if the value of <f> which applies to the given circum- 

 stances of the motion satisfied the equation (/3), it should be 

 derivable from the general integral of that equation. But this 

 is impossible, inasmuch as the equation takes no account of 

 the impressed forces. The explanation of this apparent diffi- 

 culty is, that the equation (/3), while it expresses the principle 

 of constancy of mass, has no other general integral than the 



one I have already obtained, viz. V= 5V* ^ n an y particular 



case of motion this value of V may be assumed to be the same 

 as that in a given line of motion, but only through an indefi- 

 nitely small portion of it, for which the magnitudes and positions 

 of the radii of curvature r l and r" may be assumed to be constant. 

 By treating these magnitudes and positions as variable parame- 

 ters, the coincidence of the values of V may be made to extend 

 through the whole length of the line ; and so for any other line. 

 The analytical circumstances are therefore analogous to those of 

 the general integral and particular solution of certain differen- 



