Contacts of Lines and Surfaces of the Second Ordei\ 137 



A. Quadratic loci in a riglit line. 



Simple contact.! ^y 

 One condition . _) ^-2 -)- xy . 



B. Quadratic loci in a plane. 



1st Class. 



Simple contact."\*'H?/ + ^^ 

 One condition.J ff^2^^^2_|_^2^2_ 



Proximal contact.! ^■^ + y.^ + 2/^ 

 Two cowMion?,. J ax^-^- ay x + by z. 

 2nd Class. 

 Diploidal contact.! -'^^ + 2/^ + ^" 

 Two conditions. J fl^.2^ ^^2 _^^2-^_ 



Confluent contact. ! oo" + rj- + xz 

 Three conditions._J y^^,^^ 



C. Quadi-atic loci in space. 

 1st Class. 



Simple contact.! «^ + y^ + ^H xt ") 

 One condition. J «x^ + 5j^2^ (,^2_^ ^^^J" 



Proximal contact.! «^ + 2/^ + ^^ + ^^ \ 

 Two conditions.j ffa;'2 + bif' + cxt + ast f 



Unilinear contact. ! 2 i .,,4.--/ "l 

 1st species of diploidal. > ^ ~ J> 



Two conditions. J «y^ + *-y + ^^^ -* 



Confluent-unilinear, ! ^^,2 + ,.- + .,.y + ,,.n 

 or triple contact. r ( 



Three conditions, j az^' + bxy + bst J 



2nd Class, 1st Section. 



Biplanar contact. ! _^,2 _^ 2 ^ ^2 ^ f -1 



2nd species of diploidal. >' J " ^ y 



T\\-o conditions. J ^<*'- + «!/■ + b- + c/- J 



Confluent-biplanar con- \x^ + zt-\- y^ -\- z^ ! 



tact. Three conditions. j^,p2 _j_ ^,_^ ^ j„2 _|_ f^2j 



Bilinear contact. 1 ^'^ + y^ + ^:^ -f- r^ 1 ^^^ f a'^ + yt 

 lliree conditions, f ax^ + ay'^ + bz'^ + cztj \axt + byz 

 Conflucnt-bilinearcon-! ari'— a?^ + y + ~^ 1 



tact. Four conditions. J ^^.^ ^ ^_^,^ + ^^2 ^ ^.2 / 



2nd Class, 2nd Section. 



Quadriln.ear or ! 2+ 2 + _.2 + ^2 n r.^.y + ,-/ 

 (|uadnii)U! contact. ^ ^-^ ' r""^! 



I'our coudilions. «'^'^ + «// + ^-^ 4 /</ J I- nxy + bzt 



