52 REPORT 1871. 



necessaiy to proceed beyond the first order of small quantities, tlie followiup: equa- 

 tions, expressad in the'usual notation, were adopted as being sufficiently general 

 aad approximate for the purpose : — 



dhW dx' ^ dif ^ dz- ^ ^ 



2!(^ = Xf?.r + Ydy + Zdz-d/% (2) 



p dt 



The proposed method of solving the problem of tides requires, first, that equa- 

 tion (1) should be satisfied by a particular integral of assigned form; and then 

 that the arbitrary quantities contained in this integral, together with that arising 

 from the integration of equation (2), should admit of being determined by the 

 given conditions of the problem. Before giving the details of the method it is 

 necessary to state the meanings of the literal symbols. 



The resolved parts of the velocity being u, v, iv at the point xyz at the time t, 



d(f) = udx -\- vdy + wdz. 



The attractions of the earth and moon at the unit of distance being respectively G 

 and 7)1, the impressed forces X, Y, Z are the resolved parts of the forces 



r' being the distance of the particle at xjjz from the moon. The angular distance 

 of the moon westward from the meridian of Greenwich at the time t reckoned from 

 the (xreenwich transit is ^lt. If X be the north latitude, and 6 the longitude west- 

 ward, of the point xyz distant by ;• from the earth's centre, 



a-=;-cos"A.cos^, y = rcoa\s\a.6, :=/-sinX. 



_ After transforming by these forinulaj the rectangular coordinates in equa- 

 tion (1) into the polar coordinates r, 6, \, for certain specified reasons the author 

 assumed that 



r(f> = f(r) cos^X sin 2(6- jxt), 



and then found that equation (1) is satisfied by this value of >•</> if the form 

 of/ be determined by integrating the equation 



|:C-(6-i^)/=0. 



This integration gave, after omitting the extremely small quantity ^p^, the fol- 

 lowing value of (p, containing two arbitrary constants : 



4, = (Cr- + CV-3) cos^Xsin (2d-fjit). 



The remainder of the reasoning depends altogether on this value of (j), which 

 was considered by the author to be indispensable for tlie solution of the problem 

 of atmospheric tides, and, as far as he was aware, had not beon before employed 

 for that purpose. 



For determining the three arbitrary quantities there are three conditions. That 

 introduced by the integration of equation (2) is determined by the condition 

 that at either pole of the earth the density has a constant value, because, as may 

 be inferred from the expression for </>, tlie aerial columns having their bases at 

 the poles are motionless. A second condition is, that at the earth's surface the 



vertical velocity, ^, is always zero ; so that if b be the earth's radius, C' = -— . 



The third condition necessarily has reference to the circumstances of the fluid at 

 its superior boundary, respecting which the author argues as follows : — 

 _ That the height of the atmosphere is limited may be inferred from the considera- 

 tion that, by the continual diminution of the density with tlie distance from the 

 earth's surface, the upward molecular repulsion must eventually be no greater than 

 the downward acceleration of gravity, in which case there can be no further upward 

 action, and the fluid terminates by "an abnormal degradation of the density down 



