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’ . 1 — 41) al . . . N 
Stu of Spr—i (7, 9) to Ss » where Sj now again indicates.exclusive- 
ly the spherical boundary, then the formulae of Gerpix furnish us with 
Vs = 27 2, cos a . Vay Sie = Er Lg COS . Sn | 
Vi den En Su, = 20 Es 5 yaaa 
and from this ensues immediately 
Ve — Ve cosa B Sit, == Sito cos a 
and therefore what was assumed above, so that only for « = 0 the 
proofs have vet to be given. We commence with the volume. If w 
. . a ya - ’ 
is the distance from Ss to a parallel space Ss cutting Sp,—1(”, @) 
in a spherical space Spo. with 7 =V 7? — 2’ for radius, then the 
demanded volume is 
rr 
V = 2 voe y"—2 ada 
=a 
and this passes, as #° + y? =7" and «dx + ydy = 0, into 
f 
2 
V = Ar vn—o f yr | dy = — O2 0" = Un OP, 
n 
o 
with which the special case of the theorem for the volume has 
been proved. 
In the special case of the theorem for the surface we regard the 
superficial element generated by the rotation of the surface Su, (r, ©) 
. y(t) rahe dx 7 2 
situated between the parallel spaces S,> and SEE) If ds is the 
apothema of this frustum the demanded surface is 
mr 
Su = 22 a ads, 
niel Ors 
With the help of the relations yds =rdu and ade + ydy=0 this 
passes into 
hd . 
Su = 2ar maf dy = 5 PS. = Zr. Ue "2, 
== 
o 
Le. the desired result. 
Of course we can represent to ourselves the more general segment 
of revolution Sp(7, 0, jr of order & generated by the rotation of a 
