Vol.. 8, 1922 MATHEMATICS: P. FRANKLIN 267 
From the form of these equations we see that the totality of transforma- 
tions corresponding to them forms a group ; since the identity is given by 
setting CO = 0, the inverse of the transformation co = co' by setting co = — coi, 
and the resulting of the two transformations co = coi and co = co2 by 
setting CO = coi + C02. We further note that r is restricted to quantities 
small compared with c, we may replace cosh cor/c by unity and sinh cor/e; 
by oir/c, and on neglecting the term in r'^/c'^ we obtain (2a). This justi- 
fies the use of (2a) as a first approximation to obtain results for actual 
rotating systems, for which of course r/c \s negligibly small. (Cf. the dis- 
cussion of Sagnac's experiment given by M. V. Laue, Relatimtdtstheoriey 
Vol. 1, 1919, p. 125 and by L. Silberstein in /. Optical Soc. Amer., 5, 1921, 
p. 291 f.) 
While the relation between the primed and unprimed coordinates, 
given by (9) is formally symmetrical, the corresponding spaces are 
not interchangeable. In fact, if the unprimed space is a "station- 
•ary" system the square of the arc element of the space-time continuum 
is given by : 
da'' - c^dt' - (10) 
where da is the arc element for three-dimensional Euclidean space, and 
consequently in our coordinates by 
dz^ + dr^ -f r2 ^92 _ ^2 j^2^ (H) 
The corresponding element in the rotating, primed system is obtained by 
solving (9) and substituting the values in (11). This, of course, leaves the 
four-dimensional space unchanged ; but the three dimensional section ob- 
tained by putting f equal to a constant is no longer the arc element of 
Euclidean space, but is given by the expression (in which primes are 
dropped) : 
dz^ + r^de^ H- 2f dr defeat - — sinh — cosh — - Gsinh^ —) 
\ r c c c / 
+ dr{ 1 _ ?f5:!lV + sinh^ 'iL-Q^ sinh^ ^ (12) 
\ r"^ c c 
.^.•10^^ ^ ct"^^ . , cof , cor ^rG^oo . , cor .oir\ 
— 4coG^smh2 — — 2 — smh — cosh 2 smh — cosh — )• 
c r c c c c c J 
To prove that the space with the above element is non-Euclidean, we have 
merely to show that its Gaussian curvature is not zero. We only cal- 
