of Qimternions. 493 



direction of the operand line is in general completely arbitrary 

 in its own plane ; but this is not generally true when it is a 

 line in space not confined to a determinate plane. I now pro- 

 ceed to consider the application of algebraical symbols to this 

 case. 



If we assume the symbol +^ to represent the operation of 

 turning a line through a complete revolution in a plane per- 

 pendicular to a determinate axis r, drawn from the origin, 

 then ( 4-^)*' will represent the operation of turning through an 

 angle 2a7r; and we may use — ,. as a symbol equivalent to 

 (+y) , denoting a semi-revolution. So far as rotations in this 

 one plane are concerned, all that has been said of ( + ) and 

 ( — ) in plane geometry will apply to +,. and — ^. I will add, 

 however, one remark, which may be useful to readers (if this 

 paper should meet with such) to whom the subject is new. If 

 we admit (which, however, is not necessary) the ideaof «^^«- 

 tive angles^ then, 5 representing an angle described by a de- 

 terminate rotation, —9 represents an equal angle described 

 by a contrary rotation. But if q represent the operation of 

 describing the angle 9, or the rotation by which it is described, 

 then the operation of describing —9 is represented, not by 



'^5', but by — or q~^. The equation fi + (— 5) = means the 



same thing as q~'^q-=.\. The former expresses that the an- 

 gular distance between the first and last positions of the de- 

 scribing line is ; the latter, that if we turn a line through 

 any angle and then back again, the result is equivalent to 

 simply taking it as it is, which, considered as an operatio7i, is 

 represented by the symbol 1. To justify the assertion that 

 the consideration of negative angles is not necessary, it will be 

 sufficient to observe that an angle which is negative when 

 considered as described about one axis, is positive if it be con- 

 sidered as described about an opposite axis. But a complete 

 discussion of this point would involve an examination of the 

 theory of indices, and would lead us too far from our imme- 

 diate subject, to which I return. 



For convenience let q be put for ( + ^.)'*, so that <7 is a symbol 

 of rotation, and represents the operation of turning the ope- 

 rand line through an angle 9( = 2a7r) in a plane perpendicular 

 to a given line r (whose length is immaterial) drawn from the 

 origin ; the direction of rotation, moreover, being such that 

 the line r shall be its positive or north axis. The initial po- 

 sition of the opei'and must be in the plane of the rotation, 

 otherwise the operation could not be performed upon it : but 

 ,^ far as this one operation is concerned, it may be in any 



