142 



Then determine the points R f , R" . . . R<> in which PQ cuts the first, 



and the corresponding pairs of points S/, S a ', S/', S 2 " S/" 1 ), S 2 W 



in which it cuts the second, and draw R'M/=^ 1 .PS/, 



R'M 2 '= V=\ . PS 2 ', R0> M 1 = 4/^T . PS/), 



B() M 2 0>= 4/^T . PS 2 . 



The loci of M, R, S are the principal scalar and clinant radical loci 

 of the formation /(a?, y). 



From this it is concluded, that if C^ O 2 . . . O n O l he a completely 

 enclosed polygon, the sides of which are taken in order, to replace 

 PQ, first in the direction O x O 2 , O 2 O 3 , &c., and then in the direction 

 2 O lf O 3 O 2 , &c., and the points M lf M 2 . . . M, M/, M' 2 . . . 

 M/ TO ), M 2 ( TO ) be determined for each direction and each side, and 

 [O 1>2 M-5-O 2jl M] represent the product of the ratios O 1 M-^O 2 M, 

 &c., the O l M referring to the origin O l and the M being determined 

 with reference to the direction O x O 2 , and the O 2 M referring to the 

 origin O 2 and the M being determined with reference to the direction 

 O 2 t^, and so on, then we shall have in all cases, 

 [0 1>2 M-K) 2>1 M] . [0 2 , 3 M^-0 3)2 M]....[O njl M-:-0 1>ft M] = l, 



and not = + 1, as supposed by Plucker, whose error is traced to its 

 origin, and displayed in all the examples he has given, where the roots 

 of/=0 are scalar. This result is then applied to a simple case where 

 some of the roots of /=0 are clinant, and a result obtained in ac- 

 cordance with other considerations. 



II. " A Seventh Memoir on Quantics." By ARTHUR CAYLEY, 

 Esq., F.R.S. Received February 28, 1861. 



(Abstract,) 



The present memoir relates chiefly to the theory of ternary cubics. 

 Since the date of my Third Memoir on Quantics, M. Aronhold has 

 published the continuation of his researches on ternary cubics, in the 

 memoir "Theorie der homogenen Functionen dritten Grades von 

 drei Veranderlichen," Crelle, t. Iv. pp. 97-191 (1858). He there 

 considers two derived contravariants, linear functions of the funda- 

 mental ones, and which occupy therein the position which the funda- 



