( 489 ) 
* 
ws fer dv, 
if the integral is extended to all the elements of volume of (20),41. If 
now Vj is the volume of (70),41, we can imagine a quantity , 
satisfying the equation 
fare i mrt fas =a" p—l Vo 
and we can insert this quantity in the above formula. By this it passes into 
== VA » Sn—p min. 
If we call x the “radius of inertia of order n—p—1” of the 
volume Vr, of the rotating figure (Po), with relation to the 
( L . . . Y kl . 
axial space Se lying in its space S,41, we find this theorem : 
We find the volume of the figure of revolution 
generated by the polytope (Popp rotating round 
\ 
i} 
. ct . . nies 
an axialspace S,’ not cutting this polytope of its 
space Sp, if we multiply the volume Vo of (Po) 
by the surface of a spherical space SD having 
the radius of inertia of order n—p—1 of Von 
E 9 ola) c 5 
with relation to S,’ as EO Orne 
2. Determination of surface. If in the above we 
substitute the p-dimensional element of surface for the p + 1-dimen- 
sional element of volume and in accordance with this for the volume 
Vp41 and its radius of inertia the surface Suy and its radius of 
inertia, we arrive in similar way at the theorem: 
We find the surface of the figure of revolution gener- 
ated as above if we multiply the surface Su, of (Po),41 
by the surface of aspherical space Sp,_,, having for 
radius the radius of inertia of order n—p—1 of Su 
with relation to Sa 
3. The segment of revolution. The opinions will differ 
greatly about the use of the n-dimensional extension of the Gurpin 
formulae proved above. Those regarding only their generality and 
their short enunciation may rate them too high, though reasonably 
they cannot go so far as to believe that these formulae allow the 
volume and the surface of a figure of revolution to be found when 
the common principles of the calculus leave us in the lurch, as 
the quadratures can be indicated but not effected in finite form. 
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