230 The Rev. J. Challis on a new Method of Investigating 



shaw has given of a very similar proposition in a memoir on 

 Fluid Motion, contained in the Cambridge Philosophical 

 Transactions (vol. vi. part ii. p. 204). 



Let w, v, w be the resolved parts in the directions of the 

 axes of rectangular coordinates, of the velocity V at a point 

 of the surface whose coordinates are x, y, z; and let x + dx, 

 y + dy, z + dz be the coordinates of another point of the 

 surface distant by ds from the former. Let the line ds join- 

 ing the two points make angles «, /3, y with the axes of coor- 

 dinates, and let the direction of the velocity V at the point 

 x y z make the angles a!, /3', y* with the same axes. Then 



= N {ud x ■+• v d y + w d z) 



VTr , / u dx , v dy w dz\ 

 \ V d s V ds V d s/ 

 = N V d s (cos a cos a' + cos (3 cos./3' + cos y cos y'). 



Consequently, as the factors without the brackets do not 

 vanish, the quantity within the brackets, which is the cosine 

 of the angle that the direction of motion makes with a tangent 

 to the surface, must be equal to zero. Hence the motion is 

 in the direction of a normal to the surface. 



As the differential equation N (u dx + v dy + isodz) =0, 

 may contain the time t in any manner, its integral will be of 

 the form F (<r, y t z, t) = $ (t); or, differently expressed, 

 f (x, y, z f t) = 0. For the sake of brevity I will call the sur- 

 faces given by this equation surfaces of displacement. In any 

 proposed instance of motion there will be an unlimited num- 

 ber of such surfaces, each of which will in general be conti- 

 nually varying its form and position, as well by the change of 

 t as by the change of form of the functiony depending on the 

 arbitrary circumstances of the motion. It will be a restricted 

 case, but sufficient for our purpose, to consider the change of 

 a given surface of displacement to depend only on the change 

 of t, the function f retaining its form. Then the coordinates 

 of the particle which at the time t were x, y, z, will at the 

 time t + dtbe x + u d t, y + v a It, z + <w d t. Consequently, 

 as these are coordinates of the surface of displacement in its 

 new position, 



f(x + udt, y + vdt, z + wdt, t + dt) = 0. 



Hence taking the letter f to represent f{x, y, z, i) y . we shall 

 have 



f + if udf + if vdii .if W dt+i^dt = o. 



T. dx dy ' dz dt 



But 



/=0; ^/=N«; ^=Nt;; ^=Nw. 

 dx dy dz 



