284- Prof. Challis on a nexo Equation in Hydrody7iainics. 



cP ^ il^ iP^ d^ d^ d-</> d^ d^ 



dz^' dz^ dxdy' dx' dy dxdz'dx'd?: 



+ 2 



d^j> d4> d4 {d^.d^A d^'^ 



^_(d^,d^d^\ 



dy d z ' dy ' dz \ d .r^ dy- 



(cl^ ,d£ d£\ 

 ^ \dx'''^ dy^'^ dz^l' 



Now equation (1.) is equivalent to the following, 

 dp dp dp dp du dv dw 



qdt qdx qdy qdz dx dy dz 



which, by substituting y- for u, -j^ for v, and -r- for mj, and 



putting (rfP) for the complete differential of P, or k Nap. log. q, 

 with respect to space, assumes the form 



dV (rfP) , (du (Iv ^ dw\ ^ 

 dt dt \dx dy dz/ 



It must, however, be borne in mind, that on account of the 

 preceding substitutions for u, v, and w, the variation in {d P) 

 is from one point to another i7i the line of motion. Hence 



(4Pl,o. V.^; =V.!^. Also '4? = /.. 4, and ^ 



dt Y dt ds ds qds dt 



= k . — f-. Consequently, by substitution, 



dp ^T dp (du dv , dwX ^ ._. 



Tt + '^'Ts + ^\crx'-dy-'d-.)='' ••• (^-^ 



Again, since ?< = N. ^, equation (3.) gives 



rfc/> d4 (dc}>^ d^ cr£\_ 



dlc'Tt^'' \dx' '^ dtf^ dzV ~ "' 



whence by differentiating with respect to x, 



d^(f) d<^ d^(f) f/0 

 du_ dx'^' dt d xdt ' dx 

 dx~ rfg J^ gs 



d^ (l^ /d^ d^ d^ d^(f> d4 d^ (f) \ 

 dt ' dx \d X ' dx^ dy' dxd y dz'dxdz) 



^ 7 d~^ g d^Y 



\dx^'^ dy''^ dz"") 



