302 Prof. Challis on a new Equation in Hydrodynamics. 



that X is some function of vj/ and t. Hence equation (2.) 

 becomes 



and consequently vp is a function of s and t. The equation 

 rl/ = is therefore equivalent to s=-f{t)y which equation cannot 

 be true of all the lines of motion unless they are rectilinear. 

 Hence when udx-\-vdy + wdz is an exact differential, it is 

 allowable to say that ds=dr = dr' ; and thus the equations 



V= ^J) and V= i^, 

 rr r 



applying respectively to motion in three and in two dimen- 

 sions of space, are legitimately inferred from the equation 



dY ^/l 1\_Q 

 ds \r r'/"~ 



As, however, these results have been obtained prior to the 

 consideration of any particular instance of motion, they can 

 only relate to the general law of the unconstrained action of 

 the pai'ts of the fluid on each other. In maintaining this 

 result, I am far from asserting that instances of curvilinear 

 motion may not take place under given arbitrary circum- 

 stances, when udx + vdy-{-wdz is an exact differential. 



I have now adverted to all the points of any moment con- 

 tained in Professor Tardy's communication. While I admit 

 that Professor Tardy has detected faults in my reasoning, I 

 maintain that nothing has been urged which in the slightest 

 degree invalidates the new equation; but, on the contrary, 

 that the correction of these errors, pardonable perhaps in a 

 new field of research, has only served more effectually to 

 establish the truth of that equation. 



Cambridge Observatory, 

 March 20, 1850. 



[Subsequently to the publication of our last Number in which 

 Professor Tardy's paper appeared, and therefore too late for inser- 

 tion, we received from him the following Note, which we thought it 

 riffht to communicate to Prof. Challis as soon as he informed us of 

 his intention to reply. — Ed.] 



A 



" I have taken the equation V= —, — -, without examination 



rr'drdr 



from Professor Amici's Course of Hydraulics ; but having had oc- 

 casion to consider the matter more at leisure, I have found that the 

 above expression is inadmissible." 



