260 G. H. KNIBBS. 



In homaloidal space the order of succession in the generative 

 operations is indifferent, that is to say they are subject to the law 

 of commutation. 



6. Inverse fluxional generation and its laws. — The reduction of 

 w-dimensional to (n — k) dimensional figures may be called inverse 

 generation. Its scheme may be represented by 



{[(pqrs U yzw ) < s w ] < r,} <~q y =p x (2) 



the inverse operator < — denoting that the operation is reductive. 

 It is evident therefore that : — A geometrical figure of n dimen- 

 sions can be reduced to one of (n - k) dimensions, by k successive 

 motions or displacements, of the generatrices oi n - 1, n- 2... n — k 

 dimensions in directions inversely parallel to those by which it 

 was or could be generated. 



In homaloidal space the order of reduction is indifferent : that 

 is to say the inverse operations are (also) subject to the law of 

 commutation. These two propositions are merely the obvious 

 inverse of the preceding ones. 



7. Zeros and infinities of n-dimensions. — The conception of 

 w-dimensional numerical zeros and infinities is an essential in the 

 logical use 1 of any infinitesimal calculus, and as we have seen is 

 immediately given by the consideration of spatial dimensions, as 

 illustrated in (1), § 3. Though the matter has been the subject 

 of some controversy, it is susceptible of perfectly rigorous expo- 

 sition, as may be thus demonstrated 2 : — 



1 This cannot be said of the formal use of such calculus. If a curve 

 y—f [pa) were conceived to consist of points separated a 1st order infini- 

 tesimal distance, the line joining any pair makes the angle Q— tan -1 dy/dx 

 with the axis x if the axes are orthogonal, which is sufficient for questions 

 not affecting the deviations of the curve from this direction when second 

 order infinitesimal distances are taken into account. The changes of 6 

 are dO/dx in the former case and d 2 6jdx 2 in the latter. This point will 

 be considered later. 



2 Much confusion on the conceptual character of zeros and infinities 

 from a popular impression that they cannot be distinguished, since, it is 

 thought, all relations between' quantities must disappear at the point at 

 which they vanish into nothingness, or when they transcend, as it is sup- 

 posed, our powers of representation. But as has, and will be further 

 shewn, zeros and infinities of various kinds are conceptual entities and 

 are not without conceptual properties. To meet the difficulty indicated 



