424- The Rev. Prof. Challis on the Analytical Condition of 



"Let udx + vdy + wdss be an exact differential. Then, 

 and not otherwise, it is possible to integrate this quantity, and 

 consequently its equivalent V dr, 

 from any one point of the fluid to 

 any other. P and Q (in the figure) 

 being any two points in the fluid, 

 let P R be the line of direction of 

 motion through P at a given time, 

 and let Q R represent the sur- 

 face of displacement through Qat 

 the same time. The integral of 

 u d x + v dy + iv d z, and therefore that of V d r, may be 

 taken indifferently along the line P Q, or along P R and R Q. 

 But the integral of V d r along R Q is nothing, because by 

 hypothesis this line is on a surface of displacement. There- 

 fore the integral of V d r from P to R is identical with the in- 

 tegral from P to Q. Hence if S be the integral, the differ- 



ential coefficient -7—, which is the velocity at R, is also the 



dr J ' 



velocity at Q. This reasoning applies wherever the point Q 

 is situated on the surface of displacement. Hence this surface 

 is a surface of equal velocity. Draw another surface of dis- 

 placement indefinitely near the former. Then if S-f 8 S be 

 the integral of V d r from P to r, the same will be the inte- 

 gral from P to q ; consequently, Q 5 being drawn through 

 Q in the direction of the motion at that point, we have ulti- 

 mately, 8 S = -r~ x the line Q s, and 8 S = -7— x the line Rr. 

 * dr dr 



Hence Q s, which is ultimately the interval between the sur- 

 faces of displacement at Q, is equal to Rr the interval be- 

 tween them at R. It follows that the surfaces are at all points 

 equidistant, and therefore parallel. A normal to one is there- 

 fore accurately a normal to the other, and the lines of direc- 

 tion of motion are consequently rectilinear. 



The above reasoning proves that whenever udx + vdy 

 -f w d z is an exact differential the motion is rectilinear. This 

 is the important part of the theorem I have announced, and 

 it is all that there is any occasion to contend for. In my pre- 

 ceding communication I said incorrectly that the exactness of 

 that differential is a necessary condition of rectilinear motion. 

 Nothing that I have advanced disproves the possibility of there 

 being rectilinear motion when udx + vdy + wdz is not an 

 exact differential. 



2. If u, v f w be functions of the time, and udx 4- vdy 



