472 The Rev. George Salmon on the Degree of the 



It is, I think, plain that we are not to attempt to reconcile these equations by 

 supposing the value here given for k to be erroneous ; but rather by considering 

 that the method of verification employed by me gave me the actual, as well as 

 the apparent, double points of the complex curve. 



The values of 7', t', k' for the reciprocal system are found in like manner- — 



y=2(M-2)(M-3); 3<^:=4(;u-2)(m-3)(/x-4); 

 Ik' = (a* - 3) (V - 31/^= + 77/x - 62). 



III. — Singularities of the Reciprocal Surface. 



We proceed next to determine the values of the quantities p, a, &c., for the 

 reciprocal of a given surface ; to verify that these values, substituted in the 

 equations (A) and (D), will satisfy them, and thus to show that the reciprocal 

 of the reciprocal will reduce to the degree of the given surface. We shall en- 

 deavour to determine directly as many of these singularities as we can, but we 

 are obliged to limit ourselves to the case where the original surface has no 

 multiple points or lines. We have seen that in this case the tangent cone 

 drawn to the original surface from an arbitrary point is of the degree n {n — 1 ), 



having n{n — 1) (?z — 2) cuspidal lines, and ^-^j — ^—^ double hnes. 



The reciprocal of this will be the section of the reciprocal surface by an arbi- 

 trary plane. Its degree will be 



?i=^n(n-l) \n{n-l)-\\-Zn{n-l) {n-2)-n{n-l) (n-2)(?i-3) 



= n{n-iy. 



The number of cusps in the section of the reciprocal surface, found by the or- 

 dinary rule is 



3n(n-l) jn(n-l)-2! -^n{n-\) (n-2) - 3?i(«- 1) (n-2) (n-3) 



- in{n- 1) {n- 2). 



Since, then, any section of the reciprocal surface has this number of cusps, we 

 learn that the reciprocal surface has a cuspidal line whose degree is 



c' = 4n{n — 1) {n - 2). 



