393 



" On an Application of the Theory of Scalar and Clinant Radical 

 Loci/' By ALEXANDER J. ELLIS, Esq., B.A., F.C.P.S. 

 Communicated by A. CAYLEY, Esq. Received February 20, 

 1861. Read March 14, 1861*. 



1. The following investigation, which contains a correction and 

 extension of that in Pliicker's * System der Geometric' ( 3. art. 64), 

 is a direct application of the theories in the writer's paper "On Scalar 

 and Clinant Algebraical Coordinate Geometry" (Proceedings, vol. x. 

 pp. 415-426), to which reference must be made for an explanation 

 of the notation and terminology. 



2. ~Letf(x, y) be an algebraical formation (function) of x, y and 

 of n -}- 2m dimensions, such that when any scalar value is attributed 

 to x,f=0 has n scalar (possible) and 2m clinant (imaginary) roots, 

 and let A be the coefficient of +2. Then 



3. Now let PQ be a line which, when produced, will cut the curve, 

 whose equations are 



OM=#.OI+y.OB, /(#,y)=0, 



(where x and y are scalar, and 01 is in the same straight line with 

 OP, and OB is a line equal in length to OI, and parallel to and in 

 the same direction with PQ) in the n points M ; , M 2 , . . . M w . Then, 



if OP=^.OI, where x, is scalar, ^, ^, ... Jgj* will repre- 



sent the n scalar roots ./X, A x i> . . . f n x r Hence if PQ=y t . OB, the 

 point Q not being necessarily a point in the curve, any one of the 



factors 2,,-yX will be represented by ?Q ~ r = ^ 



Now for the clinant rootsf, put y=r+i.s, where the Roman 

 1= V 1 } anc l r an d s are scalar, then must 



/(*, y) = F 1 (^, r, *) + i . *. F 2 (#, r, *), 

 because there are n scalar roots for each of which 5=0. For the 



* An abstract of this Paper has already appeared in the " Proceedings," page 141. 

 It is now printed in full by order of the Council. 



f Pliicker only says that "no imaginary point of intersection must be neglected" 

 (System, p. 45), but he gives no means for taking them into consideration. 



VOL. XI. 2 F 



