282 Prof. Challis on a new Equation in Hydrodynamics. 



dd> dd) d d> 



dx dij d z 



For by these three equations and the two preceding, ?/, ?;, to, 

 and g may be eliminated, and the resulting partial differential 

 equation in </>, x^ y, ~, and t being integrated, furnishes the 

 means of satisfying the given conditions of the motion. The 

 above are not, however, the most general expressions for 

 u, V, w ; and the partial differential equation thus obtained is 

 applicable only to limited instances of motion. I have given 

 reasons for supposing the general values to be 



-.^ dd) ., d(b ,, d<b 



d X d y d X 



N being a function in general of ^, ?/, ;rand t, the reciprocal 

 of which makes n dx + v dy +ivd z integrable. The quan- 

 tity N is given by an additional equation, hitherto, I believe, 

 unnoticed by writers on hydrodynamics, viz. 



'^t+N-l^.f .'g)=0.....(3,. 



This equation, the discussion of which is the chief object of 

 the present communication, was arrived at by the following 

 considerations : — 



First, it was shown that 



-dx + ^dy + -^ds;=0, 



is the differential equation of a surface which cuts at right an- 

 gles the directions of the motion at any given instant of the 

 fluid particles through which it passes, and which may there- 

 fore be called a surface of displacement. The integral of this 

 equation, since the left-hand side of it is equal to {d </>), is 

 ^ — 0, an arbitrary function of the time being included in <^. 

 It is plain, that as the reasoning applies to the whole of the 

 fluid in motion during the whole time of its motion, there will 

 at each instant be an unlimited number of surfaces of displace- 

 ment, differing according to different values assigned to the 

 arbitrary quantities involved in <f), and that these surfaces will 

 be continually changing their positions. Next, it was argued 

 that if X, y, ;: be coordinates of any point of a given surface of 

 displacement at the time t, x + n dt, y + vd t, z + ivd t will 

 be the coordinates of the same surface in the position, indefi- 

 nitely near the former, which it lakes at the time t + di; 

 and consequently, if when t is changed to I + dt '\n the equa- 

 tion (f> = 0, X, y, z be changed to x + u d t, y + v d i, x + "do d i, 

 that equation will still be satisfied. This consideration imme- 

 diately led to equation (3). 



