Surface reciprocal to a given one. 47 1 



m = 3(^-2), n = ^, r = 2(^-l), a = 0, ^^ = 4(;. - 3), </= i (/"- D (m- 2), 

 /, = i(V-53/. + 80), .^ = 2(^-2) (m-3), ;/ = 2(^,^-1) (/.-3). 



On substituting, then, these values in equations (E), we find 



7= 6(^-3) (,.-4); 3? = 4(^-3)(^-4)(;.-5); 

 k^{^i-Z) (2/i'-18/iH57;u-65). 



These values may be considered as so far verifying the preceding equations, 

 insomuch as it is evident, a priori., that points 7 cannot exist when ^u is less than 

 5, nor points t when n is less than 6. I may add, that I have calcidated inde- 

 pendently the order of the conditions that the equations 



At''-' + (^ _ 1) Bf'-' + &c. = 0, ^f"-' + (/i - 1 ) Ci"-' + &c. = 0, 



should have three common factors, and found the result 



(2/i-4) (2/i-5) (2m -6) 

 1.2.3 



But this is exactly the sum of the numbers §, 7, t I have similarly examined, 

 by an independent method, the number of apparent double points in the curve 

 represented by the conditions that the two equations just written should have 

 two common roots, and found the resiUt 



C)u - 2) (2m - 3) {2,1 - 5) (3m - 7) 

 1.2.3 



Now, since these conditions represent as well the cuspidal curve as the nodal 

 curve X of the developable, the number of apparent double points in the com- 

 plex curve should be h + k-\- {mx — 3/3 - 27 - a) ; the latter number being 

 that already found as representing the number of apparent common points of 

 the cuspidal and nodal curve. On proceeding, however, toi substitute the 

 values already found for h, k, &c., we find I 



( m-2)(2m-3)(2m-5)(3m-7) , ,,. . ^ .. „ , . ^^ 

 — J— 2 — 3 =^n + /c+{Tnx-S^- 2y - a) + p + y + t, 



instead of being 



= h + k+ (mx _ 3/3 - 27 - a). 



3 Q 2 



