in Fluid Media. 57 



To prove that the expression (3.) satisfies the equation (2.), it may be 

 remarked, that we readily get, by differentiating (3.) 



d?<P d 2 cj> d 2 cj> 2fx df fix (\d 2 f d 2 f d 2 f\ 

 dx 2 1 dy 2 + dz 2 ~a 3 bc dx~*~ a 3 bc \dx 2 + dy 2 + dz 2 ) 



a 3 bc \2a 2 ^2b 2 ^2c 2 / \ \dx / + \dy/ ' \dzJ ) 

 Moreover, by the same means, the last of the equations (4.) gives 



2x 

 dx ™ 2 ** 



\dx) + \dy) + \dz) ~ x 2 



^l + jL + fl Kdx/ yd y / y dz/ ~ x 2 y 2 z 2 



and d *f/Y l d*f = rf + V + -? 

 dx 2 dy 2 dz 2 x 2 y 2 z z 



which values being substituted in the second member of the preceding equa- 

 tion, evidently cause it to vanish, and we thus perceive that the value (3.) 

 satisfies the partial differential equation (2.) 



We will now endeavour so to determine the constant quantities X and 

 fx that the fluid particles may move along the surface of the ellipsoidal body 

 of which the equation is 



Ou W & 



1 = ~a T2+ ¥ 2+ J 2 (5) 



But, by differentiation, there results 



xdx ydy zdz 



a! 2 b' 2 & 2 



and as the particles must move along the surface, it is clear that the last 

 equation ought to subsist, when we change the elements doc, dy and dz in- 

 to their corresponding velocities ~~, -j- and -^. Hence, at this sur- 



(toe cifj (x& 



face, 



0-— d Jt,]Ld<P,± d JP / fil| 



~a' 2 dx^b' s dy^c' 2 dz K } 



But the expression (3.) gives generally 

 VOL. XIII. PART I. H 



