447] ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. 
239 
quadri-conical point, a 3 points each a cubi-conical point, &c., and of a simple curve 
order nil with h 2 apparent double points, a double curve order m 2 with h 2 apparent 
double points, and so on; and if moreover, the curves m 1} m 2 intersect in k lt 2 points, 
the curves m 2 , m 3 in k ij3 points, &c. ; then writing in general p = \m (m — 1) — h ; that 
is, p 1 = ^m 1 (m 1 —l) — h 1 , p 2 = \m 2 {m 2 — l) — h 2 , &c., I find that the general condition of 
equivalence is 
ax + ( 3w — 2) mi — 2pi \ 
+ 8a 2 + (12?i — 16) m 2 — 16p 2 
+ r 3 (x v + (3r 2 w — 2r 3 ) m r — 2r 3 p r I ^ . 
— oki t 2 — 8^ 3 ... — (3r — 1) ki t r 
— 28/r 2 ,3 
... —s*(3r — s) k s>r (s< r) 
sind that the general condition of postulation is 
oil + ( n + 1) Wj — pi 
+ 4a 2 + (3n +1) m 2 — 5p 2 
+ ±r(r + l)(r+ 2)a. r 
+ [|r (r + 1) n — (r + 1) (21— 5)] m r 
— Y4 [( r — 1) ( r — 2) (r — 3) (r — 4) { — £ (n + 1) (n + 2) (n + 3) — 4 : 
4- 4r (r + 1) (2r + 1)] p r 
— 2ki> 2 — 3A/j s 3... y 
8&„ 
2,3 
-1$ (s + 1) {?• 4- 1 - £ ( s + 2)} k s , r (s < r) 
in which formulae it is however assumed that the curves have not any actual multiple 
points. This implies that if any one of the curves, say m r , Ibreak up into two or 
more curves, the component curves do not intersect each other; for, of course, any 
such point of intersection would be an actual double point on the curve m r . I believe, 
however, that the formulas will extend to this case by admitting for s the value s = r; 
viz., if we suppose the curve m r to be the aggregate of the two curves m r ', m r " inter 
secting in K r points, then that the corresponding terms in the equivalence-equation are 
(3i^n — 2r 3 ) (m/ 4- m r ") — 2r 3 (p./ 4- p r ") — 2r 3 K r , 
and that those in the postulation-equation are 
r (r +1) n - fer (r -1-1) (2r — 5)] (m r ' 4- m ”) 
- ^ [( r - 1) (r - 2) (r - 3) (r - 4) + 4r (r +1) (2r + 1)] (p/ + p r ") 
— $r(r+ 1) (2r 4- 1) K r -