374 PROFESSOR CHALLIS, ON THE DIFFERENTIAL EQUATIONS 



1 1 (dp dp dpy* 

 r + ?~ [dx 7 + ~dtf + d¥) ' 



[(d*<f> d 2 (f> d*<f>\ ( d<p* dp dp\ d-(f> dp d*<f> dp dUp dp 

 \\dx* + df + dz?) [dx 1 "* dy* + ~dz 2 ) . dx^'dx 1 ~ ~df'df~ d7'dtf 



d l <p d(p dcp d*cp d<p d(p d*<p d<p d<p\ 



' dx dy' dx' dy ' ' dx dz' dx' dz ' dy dz' dy' dz)' 



and if we assume the variation in (d 2 <p) to be from one point to another 

 in the line of motion, we shall have 



dx = udt = ~- dt, 

 dy = vdt = -~r- dt, 



y dy 



dz = wdt = -~ dt. 

 dz 



Hence, by substituting these values of dx, dy, dz in the expression 

 for (d 2 <p) we obtain, 



(<F$) (Pep dp d*jp dp d?<p dp 

 dt 2 ~ dx*dx* + If' df + dz* ' dz 2 



d 2 <p d<p d(p d?<p d<p dtp d l <p d<p d(p 



' dx dy' dx' dy ' dx dz ' dx ' dz ' dy dz' dy ' dz 



Now if s be a line drawn at a given instant in the direction of the 

 motion of the particles through which it passes, and V be the velocity 



at the point xyz of this line at the time t, V = — , or dt = -= . 



Hence W> * j* && But && - u *? + v *M + w & 

 HenCe ' ~dF ~ V ' df ' ds ~ U 'ds + V 'ds + W ds' 



and the variations dx, dy, dz, being supposed to take place from one 



point to another in the line of motion, 



