104 Prof. Challis’s further investigation of the 
now = tan 4, and « = H at the upper surface of the fluid 
ie 
where p = 0, it will be found that 
U? h? 
ihe i 
Here then we have an instance in which «dw + vd y is not 
an exact differential, whether H be finite or infinitely great, 
although in the latter case the part of C which contains the 
variable sec § is an infinitesimal of the second order. ‘The 
differential of equation (13.) cannot in any case be taken ex- 
cept from point to point of a line of motion. ‘This example 
and the reasoning which preceded it, may suffice to show that 
Mr. Stokes’s argument is inadmissible. 
In my last communication I gave reasons for concluding 
that «dx + vdy + wd z is not necessarily a complete differ- 
ential when the motion is small. Another argument leading 
to the same conclusion, I find in a note at p. 464 of vol. vil. 
part 3.of the Cambridge Philosophical Transactions. This point 
may now perhaps be looked upon as settled. But the author of 
that Note contends (at p. 462.) thatuwda + vdy+wdzis 
always an exact differential when the motion commenices from 
rest. The following reasoning shows on the contrary that 
this proposition is also untenable. If ~=0, v=0, andw=0, 
when ¢ = 0, each of these quantities contains some power of 
tas a factor, and we may therefore assume that uda#@ + vdy 
4+wdz=t*(Udx+Vdy+ Wdz), one at least of the 
quantities U, V, W not vanishing when ¢ = 0. Now since 
t is unaffected by the sign of differentiation, if one of the 
quantities wd xv + vdy + wdz,and Ud« + Vdy+ Wdz 
be an exact differential, the other must be also, whatever be the 
value of ¢. But the latter quantity is not necessarily an exact 
differential when ¢ = 0; therefore ud x + vdy+ wd z is not 
necessarily an exact differential in the same case. If it be 
i and 22 each = 0, when é = O, th i 
laa ¢ da & <4 i en é = 0, the answer is, 
sec? 6. 
C=gH+ 
said that 
the identity in analytical expression of these two differential 
art ; 3 ? 1 
coefficients is not proved unless the identity of dU and av 
dy dz 
is proved, and the proof of zdentity is necessary before it can 
be concluded that the differential is exact. 
