266 
MATHEMATICS: P. FRANKLIN 
Proc. N; a. S. 
In the special theory of relativity, in place of (1) we have the Lorentz 
transformations : 
X — vt t — vx/c^ 
x' = , ^y' = y, z' = z, t' ^ -7 (3) 
To obtain the corresponding generalization of (2a) we shall assume the 
following properties, which may be used to derive (2a) from (1), as charac- 
teristic of uniform rotation. 
1. The velocity of a point in K' with respect to the point in K with which 
it momentarily coincides is independent of the time, and is the same for 
all points at a given distance from the axis of rotation. 
2. For the two concentric circles r' = r = const., the equations of 
transformation are similar to those for a uniform translation with rO the 
length of the arc along the circle in place of the linear distance x. 
3. The velocity of a point at the distance r' + Ar ' from the axis with 
respect to a point at the distance r' from the axis (both in the system K') 
is a constant (00) times Ar ' where Ar' is a differential. 
From 1 and 2 we find: 
To utilize 3, we recall the equation for compounding velocities, obtained 
by differentiating (3): {dx'/dt' = '^12; dx/dt = 1)2', v = Vi) 
•u^^ = ^2 — -^^i 
1 — ViV2/c^ 
For our case we have V2 — v{r -f Ar ) ; Vi = v{r)', V12 = coAr, giving: 
coAr = .(r + Ar) - vjr) _ 
1 — v(r + Ar) v{r)/c^ 
or, in the limit, 
dr 1 
CO 
(7) 
dv 1 — v^/c^ 
and on integration, taking the constant to make v{0) = 0 : 
V = c tanh cjot/c (8) 
By inserting this value in (4), we obtain as our final equations: 
^/ ^ i / , sinh (jor/c , 
6=0 cosh o)r/c — ct —> r = r, 
r 
/ i / ^ sinh cor/^: 
z' = z, t = t cosh Cf)r/c — r9 > 
(9) 
