( 340 ) 



(lc\('l()|)iii('iil firmi llii' Itraiii of llio Tnsoclix (»ros. it i^ cloiir lliat 

 tlie T.irsiiis hraiii slio\\> iiiiiiii>lakal>l(' ücmioüc i'clalioiis with tliis 



latter also. 



A move (lotailod accoiiiit willi li.uiire^ will Ix' |ml)lisli('(l in tiie 

 Han(ll>ii<-li <1. Eniwicklmigsgesohiolito edited l)_v IIkijtwk; 



Mathematics. — "(hi f/ir (r/iiafinn dctcnnhiinn tlw niuih'.'i of firo 

 l„>hj<lim<nisio)iriI spnrrs'. \\\ Pi'of. P. H. SfiioiTK. 



The |)rohleni which we wisii to solve is tlu' follow in,U' : 

 ''111 a s])a{'e -S, with // (linieiisioiis a reelaiiiiiilar system of coordi- 

 nates {X X., . . . X„) has heen taken and with i'(^s|iecl to this 

 system a space >), passinji' thron.uii (f has heen .uiven Ity the e(|nalions 



,i;^,_|_/= ai^j.ri + ^/o,/ ''"2 + • • • + >,'•'> ' 

 (;=1.2, . . ., r>-p) 



snpiiosing' this space S^, to have with the space of coordinates 

 () {X X., . . . Xn) hilt one i)oint (^ in common, the /> angles 

 f. f. K are to he determined between these two /MÜmensional 



spaces." 



1)V means of geometry we shonld set to work as follows. Suppose 

 in the uiven space S^, a spherical spaee having O as centre and 

 nnitv as radins and thus forming in Sj, the locus of the points at 

 distance nnit\ from (> : if this sphei-ical space projects itself on the 

 sjiace of coordinate () (A\ A', . . . X^,) a> a (piadratic space with 

 the half axes ^/,. ^/,, • . • <fp, ^vo get 



(1^ :=. cos {(^. a, = co.vr «„..... 0^, = cos (i^,. 



In an almost equally simple way the tangents of the demanded 

 anu-les are connected analytically with the central i'adii-\-ectores of 

 an other quadratic space. If /' is an arbitrary point of >), and (J 

 its i)rojectioii on the space of coordinate O (A\ A'., . . . A,,), then 

 the angle POQ^u is also determhied by the relation 



n — p 



-2" {a\i.rx -\- a2,i.r2 • • • + «/v ■v)- 

 OP' — OQ' _ /=i 



If we consider in S^, the points P the coordinates of which are 

 bound to the condition 



