1 74 Prof. P. Tardy's Observations on a 



In like manner we obtain the values of ^- and -^- ; and by 



dy dz J ^ 



adding and having regard to the known expression of-+-j, 

 and to the identity, 



c?V_rfV^^ dY_ dy^ dV_ dz__dV_ u dV v dN w 

 ds~ dx' ds "^ dy' ds^ dz ' ds~ dx'v ^~^"V^ Hz ' V' 



there results 



du dv dw_dV /I 1\ 



after which the passage from (2.) to(l.) presents no difficulty. 

 Having thus shown that the equation (6.) or (7.) is not neces- 

 sary, it remains to be examined whether it is true. 



Nor can Professor Challis say that, having employed the 

 equation (7.) in combination with that of continuity, and having 

 thus arrived at a true consequence, viz. at the equation (1.), 

 the truth of (7.) is in this manner established ; because on 

 examining his calculation, it is easy to perceive that a com- 

 pensation of errors has taken place ; and, indeed, as ^ is a 

 quantity that disappears from the final result, it was indifferent 

 to have substituted a false value of it. This will more di- 

 stinctly appear if we take the expression of — | — j, and put for 



-T-, &c. -, &c. ... , because all the terms which contain A and 

 dx k 



its differential coefficients evidently destroy each other. 



I shall begin by observing that Professor Challis ought to 



have shown that the equation 



(dv diso \ /dw _ du\ /du dv\ 

 dz dy ) \dx dz) \dy dx) ~ 



is always satisfied in order that ?/(/.r + i;<^ + W)r/^ should become 

 integrable by a factor. Moreover in his last communication 

 he explains what the equation (6.) means ; but he does not 

 assign any new and valid reason for the passage from (5.) 

 to (6.). We were already well-acquainted with that signifi- 

 cation ; but we desired to know how he could show that the 

 particles of the fluid which are on the surface 4' = must re- 

 main on it during successive instants. Nay, if the motion is 

 steady, and we add only an absolute constant in the integration 



of (4-.), and therefore take — =0, is it not evidently absurd 



to affirm that the particles of the fluid move along the surface 

 to which the directions of their motion were normal? In 

 general it is known that, in order that the particles of the fluid 

 may always remain on a surface \p(^,3/, z, 0=0j it is neces- 



