476 The Rev. Geokge Salmon on the Degree of the 



since tlie curve of contact of planes passing through a fixed point is of tlie 

 n{n— \y' degree, the number of points in which this curve meets //^ and i/ will 

 be 4?i(n — 1) (n — 2) and n{n - 1) (n — 2) (?2'— n^ + n — 12); a result per- 

 fectly agreeing with that which we otherwise obtained in the beginning of this 

 section. 



"We next proceed to determine p and a\ the number of points in which the 

 line of simple contact to the reciprocal surface meets its double and cuspidal 

 curves. This is obviously equal to the number of double or stationary tangent 

 planes which tonch the given surface along an arbitrary plane section, and is 

 therefore equal to the number of points where an arbitrary plane meets UH 

 and UJ. Hence we have 



p' = n{n-i) (n' - n^ + n - 12), a' = 4?i(w - 2). 

 The number of points {b'c') on the reciprocal surface plainly is equal to the 

 number of points of intersection of the surfaces U, H, J; hence 



{h'c') = in{n-2f (n'-7i' + n- 12). 

 Now of these points (b'c') a certain number will be stationary points [^ on the 

 curve c'. These correspond to the case where the same tangent plane touches 

 the surface along two consecutive points of the parabolic curve. But I have 

 proved already (see " Carnbridge and Dublin Mathematical Journal," vol. iii. 

 p. 44) that this will happen when at such a point a line can be drawn to meet 

 the surface in four consecutive points; and also (see "Cambridge and Dublin 

 Mathematical Journal," vol. iv. p. 260), that all such points lie on a surface S 

 of the degree llw — 24. The curve US touches the parabolic curve UH. 

 Hence the number of points in which U, S, H, intersect gives 



^' = 2n{7i-2) (llw-24). 

 Every other point (b'c' ) will be a point 7', that is to say, a stationary point on 

 the curve b'. For such a point corresponds to a plane which touches the origi- 

 nal sm-face at one point on the parabolic curve, at another on UJ. But from the 

 mere fact of the plane's touching at a point of the parabolic curve, it is a double 

 tangent plane: it must then, in two ways, belong to the system which touches 

 along the curve UJ ; or, in other words, it must be a stationary plane of that 

 system. - Hence, 



y = (b'c') - 2^' = 4n(?i - 2) [n - 3) (n' + %n - 16). 



