New Equation in Hydrodynamics, 177 



and then substitute in it for — the value deduced from (6.), 



dt \dx dy dz ) ' 



and what can be the result of this but the identity A=X ? 



If he has obtained A= — , it is because in taking the equa- 



tion (9.) for the integral of (8.), he has multiplied the last by 



1 2 



- = — . Nor has the following deduction in the last commu- 



nication of Professor Challis caused me less surprise. 



He supposes the fluid to be incompressible, in which case 

 the equation (1.) becomes 



d^ -, /I 1 \ ^ 

 and says, "since ds-=^dr=^dr^ " we have by integrating 



v=i(^i (10.) 



and if the motion take place in space of two dimensions, 



v=iW. (11.) 



r 



The assumption of ds-=.dr=dr^ is so strange, that it is need- 

 less to spend many words in order to demonstrate the incor- 

 rectness of the result. I shall only observe, that the value of 

 V may be put under the form 



rr^drdT^ 



where c?t, dr^ are the angles of contingence of the two lines 

 of curvature at the point x^ y, z of the surface of displacement, 

 and A is variable, not only with the time, but also from one line 

 of motion to another. The equation (11. )> says Professor 

 Challis, proves the proposition to which I had made some 

 objections ; but my former remarks were limited to the as- 

 sumption of equal forms for the two arbitrary functions in 

 the integral of 



and, in consequence, to the conclusion, that in the hypothesis 

 PhiU Mag, S. 3. Vol. 86. No. 242. March 1 850. N 



