172 Prof. P. Tard^^'s Observations o?i a 



original papers inserted in the Cambridge Philosophical 

 Transactions and in the Philosophical Magazine, and was 

 only aware of a brief extract of some of his results given in 

 Webster's Theory of Fluids. I have subsequently endea- 

 voured to peruse the many hydrodynaraical memoirs of Pro- 

 fessor Challis, and I am obliged to confess that I have found 

 nothing in them that could change my first opinion, or make 

 me dissent from M. Bertrand's assertions. I wish for the 

 moment merely to offer some remarks on the new equation 

 alleged to be necessary. 



The learned Professor having established by elementary 

 considerations the equation 



where p is the density of the fluid, V the velocity at the point 

 X, 3/, z, (Is the differential of the arc of the line of motion, and 

 r and r' the principal radii of curvature at the same point of 

 the surface normal to the directions of motion, has repeatedly 

 asserted that this equation could not be derived from the two 

 known equations 



dp d.pu d.pv d.pw _ .^. 



Tt^'d^^ dt/ '^'ur-^' ^'^-^ 





(3.) 



(according to the notation employed by him), and that therefore 

 another general equation was requisite, from the combination 

 of which with equation (2.) equation (1.) may result. This is 

 the onlT/ argument 1 have been able to find in the writings of 

 Professor Challis for the necessity of a third equation, and 

 he even confesses that he knows no other use for it but that of 

 deducing equation (1.). (Phil. Mag., vol. xxxiii. No. 223.) 

 Now to investigate this, he assumes that there always will 



be a factor - capable of rendering integrable the expression 



A 



udx + vdy -h wdz, so that 



u 



— t 

 \ 



The integral of this, 



4/(»,j/,2?, 1) = 0, (5.) 



where an arbitrary function of the time is included in \|/, re- 



'^dx+^d^+'^dz^id^). . . . (4.) 



AAA 



