1094 
MR. F. W. DTSOX ON THE POTENTIAL OF AN ANCHOR RING. 
§ 40. Two integrals of the equations. 
Since U is a function of z■^^ — z-^ — 2 : 3 , ^ — 23 , &c.. 
Thus S = 0 . 
Therefore 
(W cUJ 
(tz^ (tz^ 
S ?n|Cj^ — const, 
(95). 
Again, multiply the equations by z-^, Cj, z^, &c., and add. 
Therefore 
V 
Md 
27r 
(96). 
Eemembering that a^Ci = const. ; this equation gives on integration 
+ = . 
§ 41. These integrals are the equations expressing the constancy of linear 
momentum and of energy in the system. 
They are easily obtained directly. 
The momentum 
= [[ j"-^ c/<xr c /2 f/c/).(98), 
the integral being taken all over space. 
Since T' and c/^F/c/tc- are continuous everywhere, this 
= 2^7 f (Tf. - ch 
• ~ 00 
(99), 
where = value of T' at a great distance from the axis, and Tg = value of T" at 
the axis. 
Now 
_ F Ml cos (f) dcji 
" TJ" Jo\/{z — zf + — 2 nrq cos ^ 
and 
CT”?! 
{(^ - + 7.^}* 
%= 0 . 
( 100 ), 
Therefore the momentum 
