420 



the abstract (or concrete) equation respectively, in order that the 

 locus may remain unaltered.' The ordinary theory only comprehends 

 an exceedingly simple instance of the first problem. The second is 

 indeterminate so far as the representation of portions of curves is 

 concerned, so that any abstract equation may, by the help of a 

 properly selected concrete equation, represent any curve whatever. 



A curve by which the scalar value of y is exhibited corresponding 

 to any scalar value of oc in the equation $(pa, y) = 0, and which in 

 this simple case is furnished by the locus of the two equations 



OP=# . Ol+y . OJ, 0O, y)=0, 

 is termed the ' scalar radical locus ' of the abstract equation 



0O>2/)=0, 



and corresponds to what has been hitherto insufficiently designated 

 as the * locus of the equation' 0(#, y) = 0. It presents a necessarily 

 imperfect image of that equation. 



Clinant Plane Geometry. Reverting to the original pair of 



equations 



OP=# . Ol+y . OJ, $,(#, y)=0, 



and remembering that even if clinant ('impossible* or 'imaginary') 

 values were substituted for so and y in the expressions x . OI, y . O J, 

 they would still represent definite lines (see abstract of Paper on 

 * Laws of Operation, &c.,' Proceedings, vol. x. p. 85), and con- 

 sequently the line OP would still be perfectly determined, we see 

 that the limitation of as and y to scalar values in the previous inves- 

 tigations was merely a matter of convenience. We may therefore 

 give x any clinant value, and after determining the correspondent 

 clinant value of y from fy(oe, y)=0 (which will always exist if the 

 equation is algebraical), substitute these values in the concrete 

 equation, and thus find OP, and consequently the locus of the 

 equations. We may observe, however, that as a clinant involves 

 two scalars, we must have some relation given between them, directly 

 or indirectly, in order that there may be only one real variable, 

 without which limitation the locus would in every case embrace the 

 whole plane. 



The general algebraical process is as follows, the Roman letter i 

 being used for + V(-l). Let OP=P(X 1 , X 2 ...X n ) . OI, where 

 /ij ...X n =p w + i . q n , and all the p, q are scalar. We can 



