OF WHICH IS MOVING ROTATIONALLY AND PART IRROTATIONALLY. 401 
In this case 
d~X 1 dX 1 d~X 
d?^~r dr+^dd 2 
= a function of r only, and \=F(r). 
It is supposed here that the initial line from which 0 is measured is fixed. 
Let It, © be the radial and tangential velocities, so that 
xfi satisfies the equation 
R =;S=° 
dX . . 
©=— ■—=—F (r) 
dr ' 
dt dr rdo 
The auxiliary system of equations is 
dt dr dd di]r 
l ~ 0 ~-F(r) — ~0~ 
01b 
One integral is r=m, the other is , 0=n 
to 5 1 F(m) 
0 1 
Therefore the integrals are \=F(r), xjj=t-\-^—0 
The most general form of the integral will be \fi=^(m, n) where for m and n their 
values must be substituted. 
Substituting for It and © in the dynamical equations 
i’+V=f[F(r)P 
P * 
To find y, 
d X 
dt 
Therefore 
+ 
o 
dt ‘ dr rdd 
[tew* 
The auxiliary system is therefore 
dt _ dr 
T = o~ : 
dd d% 
IV) i{F(r)}*-{{F(r)}«|: 
3 F 
MDCCCLXXXIV. 
