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spherical segment Spn—1 (1.9) round a diametral space core of its 
space S,—z; of the various possible cases 
NVE GEen » n—2 
the first is the one treated above extensively. As any point generates 
at the rotation the surface of a spherical space Spy, we find — 
if along the indicated way by means of the formulae of GuLpin the 
general case of an arbitrary angle « is reduced to the special 
case a=0 — for volume Vz and the surface Sur of Sp(7, 0, Onze 
_ the formulae 
r=r 
Vik = Onkel Sk 1C08k ef ak da 
SN bee ET 
r= VP 
c=—r 
Sine Sen INS costa wba wk da 
he PAR Ee 
t=VP=-Z | 
and from this ensues the general relation 
Sun, = 290 7 cos? a Vn—» ks 
by which all cases of determination of surface except Stn,n—2 and 
Stn 3 ave deduced to simpler cases of the determination of volume. 
When determining the volume the integral gives a rational result, 
an irrational one or a transcendental one according to # being odd, 
n odd and # even, or n even and # even. And this is evidently 
likewise the case for the determination of surface. 
4. The torusgroup. By rotation of aspherical space Sp,—z(r) 
around a space So, -; of its space S, 7 at a distance a >>r from 
the centre a ring is generated in S,, the ring or “torus” 7'(7,a)n2- 
For volume Vr, a), and surface Su(r, dop of this figure of revo- 
lution of order # we find 
a 
z el 
VAO ne — SANO kel Vr—2 (a+ex)kde 
=a 
54) 
a 
8 ln 
Su (1, ane = 7 SH ni f Vr—2? (atea)Fde 
—a 
from which ensues again the formula of reduction 
Sui 27 Vion Oehoe OPENEN Dr (4) 
For the case 4=1 and k=2 the results are calculated more 
easily by means of the formulae of Gerpin, if one makes use of 
