REPORT ON CERTAIN BRANCHES OF ANALYSIS. 225 



vative signs as distinguished, from those p«^itive signs of ope- 

 ration which are used in arithmetical algebra ; but such signs, 

 though accurately defined and hmited in their use in one sci- 

 ence, will cease to be so in the other, their meaning being de- 

 pendent in symbolical algebra, in common with all other signs 

 which are used in it, upon the symbolical conditions which they 

 are required to satisfy. 



I will consider, in the first place, signs of affection, which are 

 those symbolical quantities which do not affect the magnitudes, 

 though they do affect the specific nature, of the quantities into 

 which they are incorporated. 



Of this kind are the signs + and — , when used independ- 

 ently ; or their equivalents + 1 and — 1, when considered as 

 symbolical factors ; the signs (-}- 1)" and (— 1)", or their sym- 

 bolical equivalents 



cos 2rmT-\- \/ — \ sin 2 r w tt and cos {2r -\- \) n w + 



-v/ — 1 sin (2 r + 1 ) w T ; 



2r?nr-v/~l 1 (2j-+ 1) n!r\/— 1 



or e and e^ 



The affections symbolized by the signs + 1 and — 1 admit 

 of very general interpretation consistently with the symbolical 

 conditions which they are required to satisfy,- and particularly 

 so in geometry : and it has been usual, in consequence of 

 the great facility of such interpretations, to consider all quan- 

 tities aftected by them (which are not abstract) as possible, 

 that is, as quantities possessing in all cases relations of exist- 

 ence which are expressible by those signs. It should be kept 

 in mind, however, that such interpretations are in no respect 

 distinguished from those of other algebraical signs, except in 

 the extent and clearness with which their conditions are sym- 

 bolized in the nature of things. 



The other signs of affection, different from + 1 and — 1, 

 which ai'e included in (1)" and ( — 1)", are expressible generally 

 by cos fl + v^ — 1 sin fl, or by « + /3 V^ — 1 ,where a and |3 may have 

 any values between 1 and — 1 , zero included, and where «^ -f- /3^ 

 = 1. To all quantities, whether abstract or concrete, expressed 

 by symbols affected by such signs, the common tei'm impossible 

 has been applied, in contradistinction to those possible magni- 

 tudes which are affected by the signs -|- and — only. 



If, indeed, the affections symbolized by the signs included 

 under the form cos 9 -|- -/ — 1 sin fl, admitted in no case of an in- 

 terpretation which was consistent with their symbolical condi- 

 tions, then the term impossible would be correctly npplied to 

 quantities affected by them : but in as much as the signs + and 



1 833. Q 



