axis ONX,, the plane OPE as plane OX,X,, the plane OPH as 
plane O.X,X,, then the coordinates of the points E and H are 
DEE SET CA WEU Oer ON 
TS —GSEC OP, | a —— 0 7 a0 tos tz 
and we find 
oe tn =d 
EH =asecrb — cocos p. 
From this ensues that the spherical spaces Sp,(2,ccosg++e) and 
Spn (H, a sec w-|- 9), where 9 represents an arbitrary constant, touch 
each other and that this contact is an external one or an internal 
one, according to ccosp + @ and asecy Ho having the same sign 
or not. Thus the theorem holds good: 
“If we describe out of each point £ of Qin with acos@ as z,_ 
a spherical space Sp, (2, ¢ cos y+ 9) and out of each point H of 
Oo with csect as zr, a spherical space Sp, (H, a sec p + o) 
where y represents an arbitrary constant and p and w assume all 
possible values, then two systems Syr, Sy. of spherical spaces 
Sp„ are generated with the property that each spherical space of 
one system. touches all spherical spaces of the other.” 
Both systems of spherical spaces are enveloped by the same curved 
space of order four. If namely of a rectangular system of coordinates 
with O as origin and OP as axis OX, the axes OX,, OX,, 
OX,, - - . OXppo are situated in Spy, the axes OX,, OX 43. 
ON, OX, in SS, rp, then the coordinates of two points E and 
: kde (2) (2) : : : 
H lying arbitrarily on Qi and Q,~; can be written in the form 
E H 
ct, =acosp ©, =csecw 
zt, =bsing cosg, Deh) 
rh x, — big Poos, 
ee bsin P| sin @, cos P, jj 
L, —bsingpsingp, ing, cos p, Tr) 
wr bsingpsingp, sing, Eri = 0 
SUN Pk; — 2 COS Pl-—1 
Tra bsingpsing,sinp, 49 = 0 
sin PE—2 SIN Pl 1 
zis 0 Ert3 = btg sinds, cos wp, 
aera Tha = btg W sin Wp, sinw, cos yp, 
Di 0 In—1 = big wsin yp, sinw, aes 
SIN Wn—k—3 COS Wn 9 
Zn =0 En big sin, siny,.... 
sin Wz Sin Wnr—k—e 
39% 
