80 Mr Richmond, On Automorphic Functions 



Prop. VI. The cardinal curve of a group of genus p lies upon 

 \(p — 2)(p — 3) surfaces of the second order which are linearly 

 independent. If a space of p — 4 dimensions be chosen which does 

 not contain any point of the cardinal curve, p — 3 linearly in- 

 dependent surfaces of degree three pass through the cardinal 

 curve and have the space of p — 4 dimensions as a double space ; 

 these with a surface of degree four having the space of p — 4 

 dimensions as a triple space completely define the cardinal 

 curve. 



Prop. VII. Special curves of orders ranging from 2p — 2 

 down to p 4- d — 1 inclusive always exist in space of d dimensions, 

 provided that p > d > 1 ; but exceptional cases when curves of 

 lower order exist often arise. Humbert shews that a plane curve 

 of order m is special if, and only if, it possesses an adjoint curve 

 of order m — 4. 



Prop. VIII. Every curve of the group of order m may be 

 derived from any Essential curve whose order is equal to or 

 greater than m + p by projection. The Essential curves of a group 

 are not the only curves of the group of order m = 2/x (p — 1), 

 /a = 2, 3..., in space of m—p dimensions: they are the unique 

 examples of a particular kind of curves of this order. 



Prop. IX. A non-special curve of order m must lie wholly 

 in a space of to — p or fewer dimensions. A non-special curve in 

 a plane may be of order p + 2, but not of lower order : a non- 

 special curve belonging to space of d dimensions may be of order 

 p 4- d, but not of lower order. 



A non-special plane curve of order to is always the projection 

 of a curve in space of the same order to unless m =p + 2, and 

 if m=p + 2 it is not the projection of a curve in space. 



A non-special curve of order m in space of d dimensions is 

 the projection of a curve of order m in space of d + 1 dimensions if 

 m>p + d, not if m =p + d. It is always the projection of a curve 

 of order to -+- 1 in space of d + 1 dimensions. 



Prop. X. The number of intrinsic constants pertaining to 

 the cardinal curve or to any essential curve of a group is equal 

 to the ' number of moduli' of the group = Sp — 3, if p > 1. 



The number of intrinsic constants pertaining to a non-special 

 curve of order to and genus p in space of d dimensions is 4p — 3, 

 if m =p + d. Should d be less than m —p the number is 



(4>p — 3) + (d + 1) (m — p — d). 



Prop. XI. It is clear from Prop. I. that sets of points (or 

 point-groups) upon any curve of a group may be represented and 



