of a Liquid Substratum beneath the Earth's Crust. 215 



" Taking any point on the equator as origin, let as measure 

 the longitude [linear] westward of any other point, and let 

 k be the mean depth of the water [or liquid] and y the small 

 elevation of the surface above the level at x as a definite 

 moment of time. 



" For the wave to be propagated with a persistent form 

 and at the rate a, the height at x must, in a time dt, change 

 from y to the value which y now has at a distance adt behind 

 it, or x—ctdt from the origin; that is, 



< &dt + &C.= -^-udt + &G, 



dt da; 



dt dx 



(A) 



And if this relation exist everywhere between the differential 

 coefficients, the condition will be fulfilled for finite intervals. 



" Not only the heights, but every other measure or mark of 

 disturbance must be propagated onwards at the same rate, if 

 the wave is to have a permanent character; so that if v be the 

 average forward velocity of the particles in a vertical section 

 at w, we must have 



£*-•= (*>" 



It is assumed in Mr. Heath's paper, and, I believe, usually, 

 that k, the depth of the canal, is " but a very few miles." This 

 assumption is avoided in what follows, not being compatible 

 with the problem we have to solve ; for although it will appear 

 that the assumption that k is small might have been made, 

 yet this could not have been readily foreseen. 



(3) The assumptions which will be made are : — That the 

 horizontal velocity of all the particles of water in a vertical 

 column are the same; that the vertical and horizontal veloci- 

 ties are small, and that the elevation or depression of the sur- 

 face above or below the mean level is also small ; so that pro- 

 ducts of these quantities and of their derivatives may be 

 neglected. 



From this it follows that, expressing partial differential 

 coefficients by brackets, since 



dv /dv\ . /dv' 



dv /dv\ , /dv\ 



d7 = {dT) + {dx-) v > 



= (-77 ), approximately; 



dv /dv 

 di 



