Mr. J. Cockle Ofi a new Imaginary in Algebra. 45 



were possible) be the sign of perpendicularity to the plane 

 ABC, and the transition, from regarding j as the sign of im- 

 possibility, to viewing it as the symbol of perpendicularity, is 

 by no means difficult*. As may be inferred, from what 1 said 

 in opening the question of interpretation, I am not prepared 

 to complete this view of the question in the present paper ; 

 but I cannot refrain from remarking, that j is not an unreal 

 root of unity, and that, although it may indicate perpendicu- 

 larity, yet that we must envisage it in a manner different from 

 that in which we regard i. In fact j indicates perpendicularity 

 to a plane as / does to a line; J^=l and 2^= — 1. We may 

 realize the distinction, geometrically, as follows. On the semi- 

 diameter of a sphere conceive another sphere described. Let 

 the point of contact of the spheres be considered as the pole 

 of both. Conceive two points, one at the centre of the larger 

 sphere, and the other situate anywhere on its equator ; let the 

 first point revolve in a meridian of the smaller sphere, and 

 the second in the equator of the greater, and let the angular 

 velocity of the first point be double that of the second. We 

 need only consider the relative positions of the points when 

 the first point is either at die centre of the greater sphere, or 

 at the common pole of both the spheres. It will be seen that 

 the phases (so to say) of the points correspond to those of the 

 above quantities ^'^ and i'^; and also that the first point repre- 

 sents direction perpendicular to a plane, and the second, direc- 

 tion perpendicular to a linef. I mention this because it might 

 be supposed thatj is only a second perpendicular to the primi- 

 tive axis, and, consequently, that it is only a second unreal root 

 of unity J. 



5. Of its proposed Employment in Analysis. 

 Should the admissibility of the new symbol j be established 



• The transition from unreal to impossible quantities will, perhaps, be 

 best exemplified by the two following problems: — (1.) Find three equi- 

 distant points, and, (2.), Find four equidistant points. The first may be 

 expressed by means of a quadratic with unreal roots ; the unreal quantities 

 arising from one of the points being out of the /me joining the other two. 

 So, unless I am mistaken, the solution of the last may be exhibited by 

 means of impossible quantities, which take their origin from the fourth 

 point being out oi the plane of the other three. 



\ This remark will perhaps be better expressed when we have substituted, 

 for the lesser sphere, a prolate spheroid, and then diminished indefinitely the 

 minor axis of the spheroid. Its major axis is, of course, to remain unal- 

 tered, and equal to, and coincident with, the axial semidiameter of the 

 larger sphere. 



X The symbol _/ will enter into geometrical inquiries in the following 

 manner. Suppose that we arrive at such an equation as 



then, s z=jr is the solution. 



