( 490 ) 
Others, whose attention is drawn to the fact that these formulae 
displace the difficulties of the quadratures but apparently — in this 
case displace them from definition of volume and of surface to the 
definition of radii of inertia — will on the other hand perhaps fall 
into another extreme and will deny any practical use to the formulae 
in question. Here of course the truth lies in the mean. Though it 
remains true that the GerpiN formulae help us but apparently out 
of the difficulty in the case where the direct integration falls short, 
yet by the use of those formulae many an integration is avoided 
because the radii of inertia appearing in those formulae of volume 
and surface of the figure of revolution are known from another 
source, which latter circumstance appears in the first place when 
p=n—2, thus each point P of the rotating figure describes the 
circumference of a circle and the radu of inertia relate therefore 
to the centre of gravity of volume and surface of that figure, whilst 
3 the knowledge of the common radius of inertia of 
for p=n 
mechanics gives rise to simplification. 
As simplest example of the case p=n—2 we think that a 
segment Sp (7,9) of a spherical space Sp,—1 with 7 and @ as 
radii of spherical and base boundary generates a segment of revo- 
lution Sp (7,0, @, by rotation round a diametral space SN situated 
in its space S,—, having no point in common with it and forming 
an angle « with the space S,—» of the base boundary. For this we 
find the following theorems: 
“We find the volume of the segment of revolution 
Sp(r‚e,a)n by multiplying the volume of a spherical 
space Spn with @ for radius by eos a.” 
“We find the surface of the segment of revolution 
Sp(v,e,@, Which is described by the spherical boundary 
of Sp,-i(r,9) when rotating, by multiplying the circum- 
ference of a circle with r for radius by the volume of 
the projection of the base boundary of Sp,—i(r,e) onthe 
axial space Gree 
These theorems are simple polydimensional extensions of well 
known theorems of stereometry. They can be found by direct inte- 
eration where the case «=O is considerably simpler than that of an 
arbitrary angle « And now the formulae of GuLpry teach us exactly 
io avoid the integration in the general case, showing us immediately 
that the theorems are true for the case of an arbitrary angle «, as 
soon as they are proved for « — 0. If namely w, and «ws are the 
distances from the centres of gravity of volume |, and surface 
