JoLY — On the Multi-Linear Quaternion Function. 49 



A quaternion being interpreted as representing a point, the equa- 

 tion 



r-f{pq) (11) 



establishes a relation between three points^, q, and /•. 



Let p be supposed given and constant. In this case we have the 

 general homographic transformation in space for a set of points (y) to 

 another set (r). The nature of this transformation is changed for 

 every change in the constant p ; and the equation may be taken to 

 represent what Sir Robert Ball might have called a four-system of 

 homographic transformations. The four-system of transformations is 

 more clearly exhibited by writing 



p = ^i«i + t^ttz + t^a-i + tiUi, (12) 



where the symbols a denote given and constant quaternions, while the 

 symbols t denote scalar parameters. Thus the linear transformation' 

 is 



r = tificiiq) + t.f{a.q) + t^f{a^q) + tj{aiq) ; (13) 



and it is compounded from the four given linear transformations 



f{M\ /(«2?), /(«3?), and f{a^q). 



7. In the second place, consider r to be a constant quaternion, 

 while j9 and q are variable, subject to the condition (1 1). 



The equation then represents the general space homografhy con- 

 necting two points p and §■, so that if one is given, the other is 

 generally uniquely determinate. 



Again, as in (12), we may replace r by 



r = Sxh-i + §2^2 + Ss^a + s^i^ ; (14) 



and, according to the various values assignable to the scalars «, we 

 obtain a four-system of space homographies. 



8. In the third place, \%i p = q ; then we have (Art. 3), 



''=/(??) = P(??); (15) 



and this represents the general quadratic transformation in space, so 

 that to one point q corresponds one point r, and to one point r cor- 

 respond eight points q determined by tlie intersections of thrive 

 quadric surfaces 



S;v = ^r,Y{qq) = 0, S/vr = SrjP (??) = 0, 8r,r= 8r,-p{qq) = 0, (16) 

 where Srj/- = 0, &c. are any three planes through ihe point r. 



