100 Prof. Challis’s further investigation of the 
tion of x, Consequently equations (9.) and (10.) become 
dx d? dV\ dr 
dr*drdt' \ dr) dx ™ 
adr d* adV\dr 
Getidnde * ae) dy 
0; 
. : , dt dr 
which, as the indeterminate factors 7 and Ty AY be re- 
@ y 
moved, are identical with each other and with equation (7.). 
This reasoning, which may be readily extended to motion in 
5? 
space of three dimensions, justifies the assumption in my last 
communication, that V is a function of » when udav + vdy 
+ wd sis an exact differential which may be integrated from 
any one point of the fluid to any other; and the conclusion 
thence derived also holds good, viz. that in that case the 
motion is rectilinear. 
The foregoing reasoning shows that the equations (9.) and 
(10.) are satisfied when ¢ is a function of a single variable 7, 
such that 7? = (w — a)? + (y— b?? + (2 —c)*, a, 6, andc 
being constant. To determine under what circumstances $ can 
be such a function, and the particular form of the function, 
recourse must be had to the equation of continuity. For 
incompressible fluids it is readily shown that ¢ may be a func- 
tion of whatever be the impressed forces, and that for motion 
: a reed ab) pees as 
in space of two dimensions Tp Varies inversely as 7, and for 
motion in space of three dimensions, inversely as the square 
ofr. But the equation of continuity for compressible fluids, 
when ¢ is assumed to be a function of 7 and ¢, is transformed 
into 
FA ae (IB =a ao do do do 
FMT he ~ d@ ~ “dr drdt 
dr 
2m? _ _ z— 
ota ae b) SANs a ee 
? Te di 
r r 
which does not accord in giving $ a function of 7 and ¢ unless 
the impressed force either vanishes or is a function of 7. 
It may be asked whether the supposition that ¢ is a func- 
tion of a single variable, embraces every case in which 
do  V? 
Rh oe tame is such a function. The following answer ap- 
C 
pears satisfactory. Since A and ¢ are distinct variables, the 
4+ 
