( 14 ) 



The coordinates of a common transversal of the tangents (f) and 

 ( — /) evidently satisfy the conditions 



tliorofoi'p also 



Hv eliminatiiif;- / wo lind llie e(|iiati(iii (tf llic indicated complex: 



y'l-W%2 4-y'i:, /',-. (/'i. + 'V'^n) + /':,, /'\, = '->• 



Tn lliis (-/////'c coiii/)/r.r heloii.LTs llie lijiear conjïnience /y,3i=rO, 

 p,.^ ^ 0. Its dii-ectrices / and /// arc i-c|>rcscjited l>y ,/•, z= 0, ,/•, = () 

 and ./',=::(), .r.^ = (); thc fomicr connects .1 will) llic point ('f, /^), 

 t|i(> lallci- unites A' and 0?, ft). 



Kacli ray of llic con^i-nence rests on two |)airs of tanijents; the 

 cori'cspondin^^ parameters are determined i»_v the cipiation 



/'./' + (/':. + 'V'.:.)'^ +/'.. = <»• 



So tlif' coniiilcxcone has a (hjuhh' cd^e, tlie com[)I(\\cnrve a 

 doiiitic tari'rent. 



5. This is also evident in the foUowiiiLr way. Witii ;j:i\'en vahies 

 <d' //i, //,,ƒ/:,,//, the equation /y,.^ = ;./>,, or //.,,/■, y, ,r,^ = ;.(ƒ/,./■;--ƒ/, .v,) 

 rej)resents a phuie interscctinir the coniph'xconc twice accordiiii; to 

 />i.,=(), y>j^=nO, and niorcoxcr accoi-dini;- to a I'iifht line of the |»lan(^ 



So />,, = 0, y^,., =: is a douhle e(l*:^e. 



If the plane //,,/■, — ƒ/,./•,=: A (y,,/'j — //i.'",) is to touch thecomj)lex- 

 coiie along the douhle edge, the three planes 



//:, '''i — //i '^ = '* ' .'/4 •'', — //» •'■. = <->' 

 {^\'/.-\-h^)''\ + (^^//.--'•\'M'''. + 0/4-3 ''•//.)•'■. - (^•//,4//.)-'', == '• 



must pass through one right line, so 



^■'' !l-i + ^■//4 = (>//:, , 3^ v.-, — -^^ //i = ^'//,^ 



//4 — '^^-//-i = — 9//i , -^ //, + //a = ^^!U' 



juust i)e satistied. 



By eliminating q or (> we iind 



^'' .'/: //. -h ^- (//i .'/4 - 3 //, //J + .'/, //, = 0. 



The roots of this (juadratic ecpiation determine the tangent planes 

 of the complexcone along the doultle edge, which l)ccomes a cnspidal 

 edge when 



^ 'J I Vt .'/. .'/4 = (//i lu — '^ .'/. ihY^ 



that is whcji 



.'Ji!I, = i/,yz or ^^ //, r= 9 //, y,. 



