Surface reciprocal to a given one. 47.1 



Now since the discriminant of [L'^] in general represents a number of 

 planes passing through the line joining the points .xyzw, x'y'z'vd ; this line will 

 be a multiple line in the locus represented by that equation. And in the pre- 

 sent case, where xijxw satisfies the equation of the surface, and x'y'z'w' that of 

 the tangent plane at the point, it is easy to see that this line is a double line 

 on B- — AC, and a multiple line of the degree n{n — 1)- — 6 on a . 



If, then, the arbitrary line liad been so assumed as to meet the line joining 

 xyzw, x'y'z'w', the condition R = would be satisfied even if Aj U were not a 

 factor in B- — AC. The condition that the two lines should meet {M= 0) will 

 be of the first degree in abed, a'b'c'd' ; xyzw, x'y'z'w' ; and it is plain that R 

 must be of the form M-H=0. i7 remains a function of xyziv only, and is of 

 the 4(?i — 2) degree. 



At all points then of the intersection of the surfaces U— 0, //= 0, the tan- 

 gent plane must be considered as double. H is no other than the Hessian of 

 the surface, and its intersection with U is the well-known parabolic curve, at 

 every point of which, I have elsewhere shown (" Cambridge and Dublin Mathe- 

 matical Journal," vol. iii. p. 4-1), the tangent plane touches the surface in two 

 consecutive points. 



We investigate in precisely the same way the condition that Aj 6'' should be 

 a factor in a . Eliminating between these two equations and those for an ar- 

 bitrary line, we obtain a condition of the degree n^ — 2n^ -f w — 6 in abed, in 

 a'b'c'd', in x'y'zfw', and of the degree n^ — 1n^ + li^ — IZn + 18 in xyzw. But, 

 as before, this condition must be of the form Jp'-^""*"-* /— o. /, then, is a 

 function of xyzw only, and is of the (n — 2) (n^ —«--<- n — 12) degree. We 

 learn hence that all the points of a surfxce whose tangent planes touch it also 

 at a second distinct point, lie on the intersection of the surface U with the sur- 

 face /= 0, which is of the {n—2) (n' — li' -f n — 12) degree. 



It is easy now to deduce hence the degree of the cuspidal and nodal curves 

 on the reciprocal surface. To every point on the cuspidal curve will correspond 

 a double tangent plane touching the original surface somewhere on the curve 

 UH ; and to every point on the nodal curve will correspond a plane touching 

 somewhere on the curve UJ. The points where an arbitrary plane meets the 

 multiple curves on the reciprocal surface correspond to the planes which can be 

 drawn through an arbitrary point, whose points of contact lie on H or /. And 



