162 Professor Baker, On the reduction of homograpJiy 



by {p, a'); the former is changed to the latter by the transforma- 

 tion 'S^. Let I be the line of intersection of any two planes of the 

 developable {p, a), and l' the Hne arising therefrom by the trans- 

 formation, the intersection of the two planes of {p, u') which 

 correspond to the planes of {p, g) taken. To the pencil of all possible 

 planes through I will correspond a homographic pencil of corre- 

 sponding planes through V ; and the locus of the line of intersection 

 of corresponding planes of these two axial pencils, will be another 

 quadric, say Sq. Evidently, by the construction, S is tetrahedrally 

 inscribed in Sq, in the sense explained in § 2. 



Taking then such a tetrahedron of reference as used in § 2, 

 let the lines I, V be respectively those there associated with the 

 parameters Oq and — Bq ; then the homography '^ changes the plane 

 {Oq, p), of equation 



{h - do) X +fz + vt + p [{h + 9Q)y + gz + ut] = 0, 



into the plane (— Oq,p), whose equation is obtainable by change 

 of the sign of Oq, for all values of p. When p satisfies the equation 



pQ^ + f^gu + p {fu + gv — h^) -\- fv = 0, 



the planes {6^, p), (— 6^, p) both touch S, which is a and p' . Thus 

 the plane {Oq, p) touches the quadric p, and (— 6q, p) touches a' . 



Now with the above form for the homography ^, the above 

 equation for the plane {Oq, p), which the transformation changes 

 into (— 9q, p), is, for all values of p, the same as 



{h + 9q) {a^x -f- bjy -^ c^z + djt) + / {a^x + b^y + c^z + d^t) 



+ V [a^x + b^y + c^z + d^t) + p[{h — Oq) [a^x + b^y -f c^z + d^) 



+ g {a^x + b^y + CgZ + dj.) + u {a^x + b^y + c^z + d^t)] = 0; 



the coefficients a^, b,-, ... in ^ must therefore be such that 



{h + eoja-j^+fa^ + va^ ^ {h + do Jbj^ +fbQ-b vb^ _ {h + do) c^ +/C3 + vc^ 



h-d^ ~ - f 



_ {h + Oq) di +fd^ + vd^ _{h — Oq) (22 + ga^ + ua^ _ (^ — ^0) ^2 + 9^3 +'^&4 

 ~ V ~ 



_ {h - Oq) C2 + gc^ + uc^ {h 



h + do 

 6q) d^ + gd^ + ud^ 



9 u 



and there will be no loss of generahty in supposing each of these 

 fractions unity. With the ordinary notation of matrices these 

 equations are then the same as 



h + Bq, 0, /, 



0, A-^o> 9, 

 0, 1, 



0, 

 0, 



0, 0, 



60, 



h-e, 

 0, 



h + Oq, 



f, 

 9, 



V 



u 



ds 

 d. 



