EussELL — Geometry of Stufaces derived from Cubics. 469 



aud the above equation for A : /x, becomes 

 A.2 





0. 



{ax" - 6,) {hy^ - 6,) {cz" - 6,) {dv^ - d^) {eid" - 0, 



It is unnecessary to write out the corresponding formula for the 

 inflexional tangents at P'. 



7. 27ie class and order af the Congruency . — The class of the con- 

 gruency is the number of lines joining corresponding points that lie 

 in a given plane. This number is 3, and the subject is fully discussed 

 (Salmon's " Geometry of Three Dimensions," Art. 529). 



That the order is 7 may be seen by considering the number of 

 lines joining corresponding points that pass through the point TJ on 

 the Hessian (see figure). TJ being given, so also is V, and V lies in 

 tangent plane at V \ it is in fact the point of contact of any one of 

 the six tangents from the node to the quartic section of the Hessian 

 by the tangent plane at V. The lines joining ^to the six correspon- 

 dents of the points V are six lines of the congruency, and in addition 

 there is the line TJV. 



The following is a general analytical investigation of this number 

 for any point : — 



If the line joining two corresponding points pass through a fixed 

 point x'y'z'v'oi, then 



ox = X A , 



ax 



or 



hence 



lyy' cz%' dm' 



2%f^axx' - lyy') 



ax^ + A. hy'^-\-X cz^ + A dv'^ + A eiv"^ + A ' 

 c%\axx' - lyy') - cz%'{ax'^ - hf) + ahxxjixy' - x'y) = 0, 

 dv\axx' - lyy') - dvv'iax"^ - hf) + alxyixy' - x'y) = 0, 

 ew\axx' - lyy') - eww'{ax^ - hf) + alxyixtf - x'y) = 0, 

 a;+y + z + v + w = 0. 



cz'^ax^ - Ixf) + ^ c^z'\ax^ - UfJ - '^alcxyixtj - x'y) {axx' - lyy') 



= %'{jxx^ - h/) ± \z''^{ax^ - ly'^Y - 4 "^xy[xy' - x'y) {axx' - lyy') 



