( S34 ) 



So therefore 



Vj -f In = \ n « 23 + { n « ls — \ a ls (£* — a !4 ) + constant. 



The constant is found by putting « 12 equal to « 8! = « 34 = a 19 = ^jt, 

 in which case Vj takes up the sixteenth part of the whole hyper- 

 sphere, i. e. .*■ n*, whilst Vjj becomes = 0. 



The constant then proves to he — ' 8 .t 2 , hence 



Vj+ V n =- J.V + { jt« 12 ' + \xa„— \ a ls (*ir - «„). 



Likewise we find . 

 ^Zi -f Vlll = - 'J »" + 5 w (|t - a 18 ) -f J .t « lf - 4 (4* - a l4 ) (£-t - a sl ), 

 P/W + F>K= - g »' + ' » «14 + ! » (3^ — a ls ) — 1 « 34 (irr — o ls ), etc. 



Every time the sum of the volumes of two successive tetrahedra 

 can be expressed by means of four successive elements of the first 

 cycle mentioned in § 4. We deduce easily from this: 

 Vj — Vin = \ a l9 a t4 — \a XA {\a — «„), 

 whilst in like manner we can find Vn — Viv, Vlll — Vv> etc. 

 Further we find 



Vl + VlV — K4«s« — K< (4* — O — Ks (i» — «bJ 

 and in like manner Tyy -(- IV, etc. 



If we remember that the tetrahedra I and VII are alike with 

 respect to their elements and volumes, II and VIII also, etc. and that 

 with respect to the volumes we have to deal with only a closed 

 range of six terms we see that of each arbitrary pair always either 

 the sum or the difference of (lie volumes can be expressed in a 

 simple manner. 



Mathematics. "The locus of the cusps of a threefold infinite 



linear system of plane cubics with six basepoints." By Prof. 



I\ II. SCHOUTE. 



In the generally known representation of a cubic surface S 3 on 

 a plane a to the plane sections of >S 3 correspond the cubics through 

 six points in «; here to the parabolic curve 5" of >S' S answers the 

 locus C 12 of the cusps of the linear system of those cubics. The 

 principal aim of this short study is to deduce from wellknown 

 properties of .s- 12 properties of r 12 and reversely. 



1. If a plane rotates around a right line / of >S' 3 the points of 

 intersection of that line / with the completing conic describe on / 

 an involution, the double points of which are called the asymptotic 

 points of /. According to the condition of reality of these asymptotic 



