883 PROCEEDINGS OF THE AMERICAN ACADEMY 



If Soqpo = be the equation of a cone, Snrgjo = is the equation 

 of a tangent plane. Suppose w is the line of intersection, and a, ^, and 

 y are three rectangular unit vectors ; then for different sides of the 

 cone Q = rff -^ xa, p^ = or -|- y^j, q., = xsr -\- zy, and x, y, and z are 

 eliminated by the fact that, when or -|- xa, &c., are substituted in the 

 equation of the cone, the roots of the quadratics in x, y, and z are 

 equal, because a, ^, and y are in the tangent planes. Substituting 

 rs -|- xa for q in Soq)Q ^= 0, we have 



S(or -|- xa)cf){TS -|- xa) = ^rscpts -j- 2xS«qpor -[- a;"S«qra = ; 



and, since the roots are equal, 



S^aqpw = Sor(pcr.S«q)a. 

 By introducing the other values of p, we get also 



S-|3g)or = Sorg,Br.S,3g)j3, 

 S~y(fi^ = S^cfnr.Sycpy. 

 .'. S-aqts -\- S'pfjpor -[- S'^qpor = (Sacpa -\- Spqrp -|- S/qpj') Sorqcnr. 

 But it is known that 



y a- ' b^ I c- 



Ilence we have 



/ V, S-/or S-/Gr S-Z^'or ,^ „ 

 ((jpor)2 = -~, ^, -p- = borqpV, 



r,„ S-oi'S-/or , S-a/S^/ra" , S-aJtS^/or 

 S-affa- = J L- -,, ■ ■ 



I .-, /Sn/Sa/S/ta-S/ta' , Sn/SatS/tirSyf-or , SakSaiSlTff9,!w\ 



with similar expressions for S-[3(jpor and S^j'qfor. Then remembering 

 that 



S'-ai -|- S-{3i -)- S'yi = — i^ = 1, 



SaiSaj -j- Sa;"S«^' -|~ SaZ-Swe = 0, 

 we have 



S-ai 8%/ S%1- 

 Also S«g)« =— + -^ + ~i 



. • . S«qpa + SjSqp^ + Sj-qp}' = ^ -f -^ -f -j-. 



