380 On the Expressions of Coplanarity and Homoconicism. 

 then 



V.aa,= 



(6.) 



with similar expressions the other vectors ; and consequently 



V. V. ««i .V. «3«4=(«a'i +ji/i+kz^) 





— («a?+;j/+^x) 



so that 



/3 =:aj.S.« «3a4- 

 /3^=a2.S.«,a4«5- 



-a. S.«iaga4' 

 •«j.S.a2a4a5 



i'lji/3'3/4 

 ^1» ^3>^4 



(8.) 



(9.) 



Whence, taking the product, and omitting those terms which 

 have no scalar parts, 



— S . « a^a^.S. oCj^agU^ . S . ol^ol^o.^ . S . OL^tt-^a. 



+ S . « «i«3 . S . «i«3a4 . S . OLc^OLi^OL^ . S . ag^s* 



— S .a UqUq . S . «i«3«4 . S .ct^u^a^ . S . otQCtcfii 



which vanishes identically whenever a coincides with any of 

 the vectors a^ ... a^; so that these last five vectors lie on the 

 cone represented by 



when a;, y, z alone are considered as variable. The only 

 case, which is not at once obvious, is « = «4; suppressing the 

 common factors, (9.) then becomes 



y^y^^ya 



Zj 2^ Zq 



yi^y^yb\ myayy 



^V ^i ^51 



l^> ^3» ^1 



y^^y^ysl 



y^y^y-i 



2, SjjSg 



^3> ''^j ^5 

 2^3> ^> ^5 



(or writing 



(10.) 



^=«^5-^5^J ^-^h-^bV)^ ' (11-) 



a;*! 



^i» 



i^2» 



3'3» 



. (Xori + fAj/i+vari), -(A^a + Z^yaH-"^?)* (^3 + B's + "^a) 



= (A^ + jt*y + y^) 



«^i> ^^i ^3 — 



y\'>yz'>yz\ 



(12.) 



