352 Mr. John Imray on the [May 21, 



tigation it is shown that this must be the case), the author has, in the 

 first instance, assumed that the upper origin is somewhat in advance of 

 the lower, the amount of such advance being a quantity of the order Aft, 

 which he has taken as mAh (it being afterwards proved that m= 0). 



"With this nomenclature, the equation of continuity is deduced in the 

 following terms : 



a 



constant area independent of t. 

 The pressure p at the lower left-hand angle of the element being a 

 function of Ti and , equations are deduced giving values for the horizontal 



cPx 

 accelerating force, -, and the vertical accelerating force including 



gravity, g + -^, in terms of the differential coefficients of #, y, and p. 



From these equations it is shown that -|- = 0, or that the pressure 



ctt> 



along any wave-stratum is uniform ; and this result leads to the simpli- 

 fication of the differential equations. 



From the integration of those equations it is shown that every mole- 

 cule of the liquid revolves with uniform velocity, and with the same 

 angular velocity at all depths, in a truly circular orbit, the radius of 

 which depends on the depth of the molecule below the surface of the 

 liquid. The law of variation of the radius is, that while the depths 

 increase in arithmetical progression, the radius diminishes in geometrical 

 progression, or that the logarithm of the reciprocal of the radius is 

 directly proportional to the depth of the centre. 



The resultant of the forces acting on a molecule is shown to be always 

 normal to the profile of the wave-surface of which the molecule forms a 

 part, such resultant being compounded of gravity, a constant force 

 acting vertically downwards, and of the centrifugal force of the molecule, 

 also a constant force acting radially outwards from the centre of the orbit. 

 The direction and magnitude of this resultant are represented by the 

 position and length of a line drawn from any point in the orbit to a fixed 

 point in the vertical line passing through the centre of the orbit. The 

 liquid element in traversing its circular path varies in width and in 

 height to suit the varying direction of the forces acting on it, its greater 

 height giving a greater hydrostatic pressure at the upper part of the 

 orbit, where the centrifugal force is opposed to gravity, and its less 

 height giving a less hydrostatic pressure at the lower part of the orbit, 

 where the centrifugal force acts along with gravity. Thus the uni- 

 formity of pressure throughout the orbit is maintained. 



As a molecule revolves uniformly round the centre of its orbit, this 

 centre is the mean centre of gravity of the molecule during a complete 



