( 488 ) 
on Si as centre, PQ as radius; so it can be represented by the symbol 
Spr—p (Q, PQ). 4 
The question with which we shall occupy ourselves is as follows: 
“How do we determine volume and surface of 
the figure of revolution generated by (Po)Aq ro- 
re a 
» if we assume that opp and Sy 
though lying in the same space S,4; have no points 
; (a 
tating round § 
in common?” 
This theorem is solved with the aid of a simple extension of the 
well known formulae of GurpiN, which serve in our space to deter- 
mine the volume an the surface of a figure of revolution. To prove 
these generalized formulae we have but to know that the surface of 
the above-mentioned spherical space Sp,—p (@, PQ) is found by multi- 
plying PQr-P—! by a coefficient s,—, only dependent on n—p; for 
its application however it is desirable to know not only this coeffi- 
cient of surface s,—, but also the coefficient of volume wv, by which 
PQ» must be multiplied to arrive at the volume of the same 
spherical space. To this end we mention beforehand — as is learned 
by direct integration — that between these coefficients the recurrent 
relations 
2a 27 
Un == Vn—2 A = 5 Sn? eae you oe (IL) 
1 Med 
exist, whilst the well known relation between volume and surface 
leads in a simpler way still to the equation 
1 
RB oes Oph cao oro ((24)) 
VW 
Un = 
In this way we find as far as and inclusive of m= 12 out of the 
well known values of »,, v7, and s,, s, 
] | | d | k 
| | 
dE 3 4 NG 7 8 9 10 | 11 |) 42 
| | | | | | | 
| 4 Nel Sr ae ee ee eral eee se CN 
el KE: Di eae |) GO) OAS Oo O0 [0395 | 720 ~ 
EJ y >i g 
Sn ars Ax 97x? © me 7 ze ie : a ie ic” ee rsa J xf 
| 3 15 3 105 | 42 945 | 69 
L Determination of volume. If « indicates the length 
of the radius PQ and the differential dr the p + 1-dimensional 
volume-element, lying immediately round ?, of the rotating polytope 
(Po), then the demanded volume is 
