Surface reciprocal to a given one. 479 



mined the nature of the curve of contact of double tangent planes to the sur- 

 face. By this method it would be necessary to examine when the coefEcient 

 of the cube of A^U in the discriminant of [f/] (which is of the form 

 E {B^ — AC) -^-Fa ), vanishes, or becomes divisible by A^U. I have not suc- 

 ceeded, however, in deriving the theory of triple tangent planes in tliis way. 



POSTSCRIPT— (Added Jan. 5, 1857.) 



An interesting application of the preceding theory may be made to the class 

 of ruled surfaces, which is obtained by eliminating t between the equations 



Af-\- Br' + Ct°'' &c. = 0, A'e + B't'-' + C't"'' + &c. = 0, 

 where A, B, &c., are functions of the co-ordinates of the first degree. 



Mr. Cayley has proved (" Cambridge and Dublin Mathematical Journal," 

 vol. vii. p. 171) that the reciprocal of every ruled surface is a surface of the 

 same degree. In fact, since every tangent plane contains a generating right 

 line, the number of tangent planes which can be drawn through an arbitrary 

 right line is the same as the number of generating right lines which meet the 

 arbitrary right line. Now if a + b = /x, tlie degree of the surface we are now 

 studying is fx, and it is proved by the methods which I have employed (" Quar. 

 Jour, of Mathematics," vol. i. p. 252) that the surface contains a double line 



„, , (/.-l)(|u-2) ... , (/i-2) (/i-3) (m-4) 

 of the degree ^^ ^^ ' ' °^ which there are ^^ ^ \^ — ^—5^^- -'' 



triple points. The number of apparent double points of this line investigated 



, , .1 ;, /(m-1) (^-2) (/x-3) (3^1-8) ^ ^, 

 by the same methods came out ^^= '—^ — j — ~^ — . -, but I have 



reason to believe (see p. 471) that this number includes the triple points ; 

 wherefore, subtracting theirnumber, as previouslydetermined, wehave remaining 



^J^ '--^ — n \ for the true number of apparent double points. 



And it will be found that these values agree with the two following equations, 

 derived from equations A and D (p. 465), 



(b-a) {n-2) = ?,t-K, {2b - a) {n - 2) (»i - 3) = 8A; - 28 ; 



3r 2 



