421 



reduce this expression to OP=(P 1 +i P 2 ) 01, where P l and P 2 are 

 scalar 'formations' (or ' functions') of the 2n scalars^9 r . p n , q l . .q n . 

 As this is equivalent to OP=P 1 . OI + P 2 'OJ> it is precisely the 

 same as the general scalar concrete equation lately investigated. 

 But one abstract equation will now no longer suffice ; for if we put 

 P 1 =o?, P 2 =y, we must have 2n 1 additional equations, in order 

 ultimately to find f(x y y) = Q by eliminating 2n variables between 

 2+ 1 equations. The result, OP=# . OI +y . OJ, f(x, y)=0, with 

 the conditions of scalarity, will then enable us to determine the locus 

 by the usual process. 



If there be only two clinants, X==p-f i . q, and Y = r+i . s, and we 

 have given OP=P . (X, Y) . OI ; C . (X, Y)=0, where C=0 may 

 be termed the ' curve equation/ these reduce to 



OP = P 1 .OI + P 2 .OJ, C^O, C 2 =0, 



where P x , P 2 , C^ C 2 are formations of p, q, r, s, so that, on putting 

 Pj=o7, P 2 =y, we have only four equations, between which we cannot 

 eliminate p, q, r, s. This again shows the necessity of some ad- 

 ditional relation, A.(p, q, r, s) =0, which may be called the ' assignant 

 equation,' in order finally to discover /(#, y) = 0, and thus determine 

 the locus. 



The only case ordinarily considered is where X is scalar. This 

 corresponds to putting q=Q for the assignant equation. Hence q 

 disappears and we have 



OP=P 1 .OI + P 2 .OJ, C T = 0, C 2 = 0, 



where P T , P 2 , C^ C 2 are formations of p, r, s, and hence, putting 

 P 1= =#, P 2 =y, we immediately eliminate p, r, s t and determine the 

 locus. 



This general theory is illustrated by numerous examples, and in 

 particular Pluck er's * involutions ' by means of ' imaginary lines' 

 are fully explained by help of the really existent lines of this theory. 



The general theory of the intersection of two *clinant loci' is 

 precisely analogous to that of two scalar loci with general concrete 

 equations. In the particular case where the concrete equations are 

 the same for both, or the reduced equations are 



OP^CP^, q, r, ) + i . P 2 <>, q, r, )] . OI, . 



Vi(P> Q> r > *) = 0, C 2 (p, q, r, *) = 0, A(p, q, r, s) = 0, 

 and OP'KW* '. ^ ')+i - W> ?', r' t *')] .OI, 



C/V, ?', r', *') = 0, C 2 V> q' } r', ^) = 0, A'(p', ?', r', *') = 0, 



