98 Prof, Challis’s further investigation of the 
the axes of coordinates, those equations, it is well known, are, 
cP _pX=0;. siHNlT od tae (1) 
Ae 
BP yack bainidyy, 228°, aA) CORD 94 Py 
= —-pZL=0. USTs Ob) ew (3.) 
By multiplying (1.) by dz, (2.) by dy, and (3.) by dz, and 
adding, we obtain, 
ages X)do4 (2 y) da 5 Gee Z)dz=0. (4.) 
dz ? dy ° EA ne 2 : 
And if dy = mdx, dz =nd2, this equation becomes 
dp ) (Ge (6 P ) es 
(Gh —-X + oe PY) m+ Tore & nm = 0. 
It is plain that by reason of the equations (1.), (2.), (3.), this 
last equation is satisfied whatever be m and m, and conse- 
quently that the increments da, dy, dz are in no way related 
to each other. On this account, when the right-hand side of 
the equation 
CP) = Xde + Vdy + Zdx 
is an exact differential, the integral may be taken from any 
one point of the fluid to any other. I am not aware that the 
specific reason for its being allowable to integrate between 
limits entirely arbitrary, has ever been in this manner referred 
to the equations (1.), (2.), and (3.). 
Let us now turn to the hydrodynamical equations. For the 
sake of having as simple analytical expressions as possible, I 
shall confine myself to the equations of motion parallel to the 
plane of « y, these being sufficient for carrying on the argu- 
ment. We shall thus have the two equations, 
dp lie du du dt 
aile Sit ge mE cans OF!) Ue 
dp dv dv dv 
a es sulle aneina hate » + (6) 
By multiplying (5.) by d 2 and (6.) by dy, and adding, an 
equation is obtained not generally integrable, but which be- 
comes integrable by supposing ud a + vd y to be an exact dif- 
