( 341 ) 



(•(»iilaiiiiii,ii' only .i\, .r., r^, .nul lliii^ (v\|)i'('ssiiiLi' lliul llic projcclioii 



(^ ot' P on <^{X^ A', . . . A'^,) roinaiiis in this latler s|);u'e on the 

 quadratic' sj)ai*c represented by (J), then tlie relation liolds good 



1 

 ft/ a T=^ — . 

 OR 



It" />j, />,,, . . . h^, are the halt" axes ot' this new (piadralic spaee, we 

 shall lind 



1 1 1 



f,, a^ = -J;, a.^ = - t;,(C^, = J-. 



Now (1) passes into the syniholic l"()i'ni 

 by the substitutions 



II — ;' n — 7» 



-^ fi^U = Ai,j. , ^ (q.j au — A/,.j; 

 i=\ !=\ 



SO the well Ivuow secnlai- equation 



A^ A',x I 





= 



furnishes l)v its roots ?.^, ).-^. . . . X,, the coeflieients ot" tlie (Mpialiou 

 of that (|uadratie space reduced on the axes. 

 From the relations 



1 1 1 



ensues immediately that the demanded e((ua(ioii is arrived at by 

 repiacinu' in the abox'e meiilioucd dclcrminant >. by hf- h. 



Mathematics. - " Tin' Incus of fhc prii/rip/fl (i.n's of (I pt'/ir/'l of 

 quiidnitic siir/(ici's\ \\\ Prot". .1. I'ardinaal. 



1. The envclo|)e of the axes of a |)encil of conies was imestigated 

 among othei-s by ]\I. TREiUTsruKR' ). lie found Ihallhe axes of the above 

 mentioned cf)nics envelo|) a ciu-nc of class three lonching the i-igiil 

 line at infinity of the ])lane in two points conjugate to the directions 

 of the axes of the two |)arabolae of the pencil with i-es|)ecl to tiie 



') Ueher Boziolmiiiïon zwisc.lion Kogo]sclinilll)iiscliehi un<l ralionalcii (.lnrvcii drillci' 

 Glasse, Silzungsbcr. der Kaiseil. Akademic der Wisscnschaflen, Bnd. LXXXI, p. 1080. 



