REPORT ON CERTAIN BRANCHES OF ANALYSIS. 297 



of Harriott, the significant terms forming one membei', and 

 zero the other, are said to be resolvible into simple or quadratic 

 factors." It is only another form of the same proposition to 

 say, " that every equation has as many roots as it has dimen- 

 sions, and no more; those roots being either real* or ima- 

 ginary ;" that is, being quantities which are expressible by 

 symbols denoting real magnitudes affected by such signs as 

 are recognised in algebra. 



We have before said that it is impossible to assign before- 

 hand an absolute limit to the possible existence of signs of 

 affection different from those which are involved in the sym- 

 bolical values of (1)" and ( — 1)"; and when it is said that every 

 equation is resolvable into factors of the form a; — a, we presume 

 that a is either a real magnitude, or of the form a + (3 v' — 1, 

 where « and jS are real magnitudes. If we should fail in esta- 

 blishing this proposition, it would by no means necessarily fol- 

 low that there might not exist other forms of factors like x — a, 

 where a denoted a real magnitude affected by some unknown 

 sign different from +, — , or cos fi + V — 1 sin 9, which might 

 satisfy the required conditions : at the same time its demonstra- 

 tion will show that our recognised signs are competent to de- 

 note all the affections of magnitude which are subject to any 

 conditions which are reducible to the form of an equation. 



If we assume in the first instance the composition of equa- 

 tions to be such as we have stated in the enunciation of the 

 fundamental proposition, we can at once ascertain the composi- 

 tion of the several coefficients of the powers of x in the equa- 

 tion 



and we can complete the investigation of all those general pro- 

 perties of equations which such an hypothesis would lead to. 

 All such conclusions, when established upon such a foundation, 

 are conditional only. It is not expedient, however, to make 

 the fate of any number of propositions, however consistent with 

 each other, and however unquestionable their truth may appear 

 to be from indirect or from a posteriori considerations, depend- 

 ent upon an hypothesis, when it is possible to convert this hypo- 

 thesis into a necessary symbolical truth. Using such an hypo- 

 thesis, therefore, as a suggestion merely, let us propose the 



* It is convenient in the theory of equations, for the purpose of avoiding 

 repetition, to consider symbols denoting arithmetical magnitudes and affected 

 with the signs + or — , unreal; and quantities denoted by symbols affecttd 

 with the" sign cos + V— l sin 6, as imaginary. 



