The Rev. Professor Challis in Reply to Mr. Stokes. 423 



turning to Camb. Trans., vol. vi. p. 180, it will be seen that 

 I have proved them to be unequal. I am truly sorry that this 

 misprint, or mis-transcription, or whatever it may be, has 

 caused so much trouble. It was very natural that it should 

 mislead Mr. Earnshaw, and produce the argument at p. 342 

 of Nov.' Phil. Mag. ; but I should have hardly imagined it 

 possible to have deceived Mr. O'Brien, who appears to have 

 perceived (see his P.S. p. 34:3) that I supposed the axis ofy 

 to be in the direction of transmission. 



For having given these gentlemen the trouble of arguing 

 the incorrectness of equations which are undoubtedly erro- 

 neous (if u is not n x in the last line of p. 162), I hope they 

 will accept my apology. 



I am, dear Sir, with great respect, 

 Your obliged Servant, 



Edinburgh, Nov. 2, 1842. P. KELLAND. 



LXXV. On the Analytical Condition of Rectilinear Fluid Mo- 

 tion, in Reply to Mr. Stokes's Remarks. By the Rev. J. 

 Challis, M.A., Plumian Professor of Astronomy in the Uni- 

 versity of Cambridge*. 



TV/I R. STOKES has brought forward four arguments against 

 - L " J ' a new theorem in hydrodynamics which I have advanced, 

 viz. that fluid motion is rectilinear whenever udx + vdy+wdz 

 is an exact differential. The observations I am about to make 

 in reply will follow the order of the arguments. 



1. In the first argument (p. 297) it is contended that my de- 

 monstration in the August Number of this Journal is deficient 

 in generality, because it takes no account of the curvature of 

 the lines of motion. I admit the validity of this objection. The 

 geometrical reasoning I have there given proves only that 

 u dx + vdy + iadz is an exact differential when the motion 

 is rectilinear, if the surfaces of displacement are surfaces of 

 equal velocity. I have not proved, as Mr. Stokes asserts, that 

 for the case of rectilinear motion the surfaces of displacement 

 are surfaces of equal velocity. This is not necessarily the 

 case unless udx + vdy + wd she an exact differential. 



The following demonstration derived from the equation 

 udx + vdy + isodz = V dr, is more to the purpose. In 

 this equation V is the velocity at a point whose coordinates 

 are x, y, z at a given time ; u, v, to are the components of V 

 in the directions of the axes of coordinates ; and d r is the 

 increment of space in the direction of the motion through the 

 point xyss. The proof of the equation is sufficiently well 

 known. 



* Communicated by the Author. 



