380 PROFESSOR CHALLIS, ON THE DIFFERENTIAL EQUATIONS 



being by hypothesis integrable, is the differential equation of a surface of 

 displacement. The integral of this equation, since the left-hand side of it 

 is equal to {d<p), is (p = 0, an arbitrary function of the time being in- 

 cluded in (p. As this reasoning applies to the whole fluid during the 

 whole time of its motion, there will at each instant be an unlimited 

 number of surfaces of displacement differing according to different values 

 assigned to the arbitrary parameters involved in <p; also, as <p contains 

 the time t in any arbitrary manner, these surfaces may be supposed to 

 be continually changing their forms and positions. Consequently if x, y, %, 

 be the co-ordinates of a given surface of displacement at the time t, 

 x + udt, y + vdt, z + wdt, will be the co-ordinates of the same surface 

 in the form and position which it takes at the time t + dt, the change 

 of form and position being supposed to be indefinitely small. If there- 

 fore t be changed to t + dt, and x, y, z, be changed to x + udt, y + vdt, 

 z + wdt, in the equation cp = 0, that equation will still be satisfied. Hence, 



<p (x + udt, y + vdt, z + wdt, t + dt) = 0, 



and expanding to the first powers of dt, 



<b + / udt + -$■ vdt + -£ wdt + -~ dt = 0, 

 T dx dy dz dt 



which equation, since = 0, becomes, 



ft + S**+S*.+ 'S«-0 (12). 



dt dx dy dz 



Now substituting N -4^- for u, N—^- for v, and N -r- for w, we obtain 



the equation sought, viz. 



f t +*m+ d $+m=° <->■ 



9. If in equation (12) -=y be substituted for u, -j- for v, and -j- for w, 

 the result may be put under the form -j- + T. = 0. But it must be 



