141 



March 14, 1861. 



Major-General SABINE, R.A., Vice-President and Treasurer, 

 in the Chair. 



The following communications were read : 



I. "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 ARTHUR CAYLEY, Esq. Re- 

 ceived February 20, 1861. 



(Abstract.) 



This investigation is in correction and extension of PI ticker's 

 theory of transversals (System der Geometric, 3, art. 64), and is 

 founded on the theories explained in the f Proceedings,' vol. x. 

 pp. 415-426. 



It is shown that if/(,r, y) be an algebraical formation (function) of 

 n + 2m dimensions, such that when x is scalar (possible) /=0 has n 

 scalar and 2m clinant (imaginary) roots, and X is the coefficient of 

 yn+2m j the value off(x lt y L ) may be represented geometrically by 



\ i 



" OB ' OB " OB OB ' OB ' OB OB ' 



where O is the origin, OP=^ . OI, PQ=y t . OB, and PQ is any 



straight line, cutting the curve whose equations are 



OM=*.OI+y . OB, A*,y)=0, 



(where x, y are scalar, OI is in the direction OP, OB is of the same 

 length as OI and in the direction PQ) in the n points M^ M a . . M,, 

 and where M' lf M' 2 . . . M/" , M a ( " are determined as follows. 



Put y=r+ */ 1 .s, where r and s are scalar, reduce f(x, y) to 

 the form F L (^, r, s) -f V/^I . s n . F 2 (x, r, s), put F^O, F i =0, from 

 which equations find by elimination F 3 (#, r) = 0, F t (^, *)=0, and 

 construct the loci of R and S, where 



OR=.r . Ol + r . OB, F 3 (x, r) = 0, 



OS=x . Ol + s . OB, F 4 (x, )=0. 



VOL. XI. M 



