28 Prof. Challis on a Mathematical Theory of Tides. 



tial equations. The latter, although it gives the solution of 

 the proposed problem, is not directly obtainable by integrating 

 the given differential equation, but may be deduced by varying 

 the parameters of the general integral. It is thus that in every 

 case of the solution of a hydrodynamical problem by a par- 

 ticular and definite integral, the general equation V= ~~ is 



satistied at each point, if only that integral satisfies the condi- 

 tion of making udx-Yvdy-\-wdz an exact differential. In the in- 

 stance before us it has been shown that this condition is satis- 

 fied ; and we may therefore conclude that our integral is in ac- 

 cordance with the principle of constancy of mass expressed by 

 the equation (/3), which is all that exact reasoning demands. 

 But because the integral is not obtainable by direct integration 

 of that equation, I have attached a dash to the symbol $ for 

 the sake of distinction. The views here stated have long been 

 advocated by me in my hydrodynamical researches, but they 

 have not, I think, been so clearly exemplified in any previous 

 problem. 



The following conclusions are deducible from the foregoing- 

 values of u', v', and w ! . The sign of the vertical velocity u 1 is the 

 same as that of sin 2(6 — fit). Hence, reckoning westward from 

 the point to which the moon is vertical, the waters are rising 

 from 0° to 90° of longitude and from 180° to 270°, and falling 

 from 90° to 180° and from 270° to 360°. Hence it is high 

 water along the meridian under the moon, and along the opposite 

 meridian, and low water along the meridians 90° distant from 

 these. 



The moon being supposed to move in the plane of the earth's 

 equator, the tide on a given meridian varies as the square of the 

 cosine of latitude. 



The horizontal velocity in longitude (v) is negative, and 

 therefore eastward, at the meridian under the moon, and at the 

 opposite meridian, and westward at the meridians 90° from these. 

 At the four meridians midway between these four it vanishes. 

 This velocity varies on a given meridian simply as the cosine of 

 the latitude. 



The horizontal velocity in latitude (w 1 ) is in directions from 

 the equator where the tide is rising, and in directions towards 

 the equator where the tide is falling. It vanishes at the equator 

 and the poles, and has its greatest value in the north and south 

 latitudes of 45°. 



The ratio of w' to v is equal to — sin X tan 2(6— /it). 



Generally, the tides consist of two great waves in opposite he- 

 mispheres, stretching from pole to pole, and travelling round the 



