OF MAGNITUDE AND DIRECTION. 281 



into — P while the angle of direction varies from to two right angles, that is, from the orio-inal 



direction to the exact opposite. The answer given by the preceding investigation is, that the symbol 



e 

 (- l)" is that by which P is to be affected; the equivalence of this symbol to cos + (— l)* sin 

 is a proposition of pure analysis, and it must therefore be allowed that cos 9 + {- \)i sin 6 is the 



fl 

 true sign of affection for P, if it be granted that (- ly^ is the correct sign of affection; in fact, 



B 



the symbol (— l)" expresses the continuous passage of a quantity from + to — , or of an affection 

 to the exact opposite, it is an algebraical fact that the symbol cos ^ + (— i)5 sin 9 will express 

 the same. If therefore there be any flaw in the reasoning it must be in that part by which the 



form of the symbol (- 1)" is established; but on reviewing that part it will be seen, that the only 

 assumption is that an oblique cause can be symbolized by an expression of the form Pf{9), where 

 P is the magnitude of the direct cause; for if it be granted that Pf{_9) is the expression for 

 an angle 9, it seems evident that P\f{ff)\^ will be the expression when the angle is doubled, 

 since P\f(9)\'' is derived from Pf(6) according to the same rule by which Pf{9) is derived from 

 P; the only question therefore is as to the legitimacy of the assumption of the symbol Pf(0), 

 and this is, I think, justified by the consideration that the expression for the oblique cause, if 

 there be one, can involve no other quantity of the same kind as P, and therefore P must occur 

 as a factor of the expression; the form could not, for instance, be Pf{9) + <p(9), since <p{9) 

 would be a quantity incommensurable with P. But if this step be granted, the conclusion that 



f(9) = (- 1)" seems inevitable, and the equation f(9) = cos9 + (- l)' sin 9 is consequently true, 

 being algebraically deducible from the former. 



8. Supposing then the demonstration free from real objection, we may regard the formula 



e 

 f{e) = (-1)- = cos9 + (-l)i sine (A) 



as expressing the law, according to which a cause depending solely on magnitude and direction 



w 



ill vary to its exact opposite, a law true in the nature of things and which will only require 

 to be interpreted in the case of different causes which come under the definition ; the question 

 will arise, can the formula be interpreted in a manner free from objection, at least, can an 

 interpretation which can be depended upon, be put on a symbolical formula from d priori 

 considerations ? I have already transformed the equation (A) into the equation (B), and have 

 made use of that form of the equation to give this interpretation, that the effect of an oblique cause 

 is measured by that of two whose directions are at right angles to each other, the direction of 

 one being the original direction that of the other the perpendicular to it, and the intensity of 

 the former being measured by P cos 9 that of the latter by P sin 9. Now perhaps this inter- 

 pretation may appear doubtful, when we consider that the .sign + connects two quantities which 

 express causes acting not in the same direction, and whose effect cannot therefore be summed 

 according to the usual rule; but I think this difficulty will disappear when we consider that the 

 possibility of resolution of a cause into two components is an obvious truth which may he seen 

 independently, that is, it is in general possible to find two causes in given directions, whose 

 effect shall be the same as that of a given cause, and the only peculiarity of the case in whicli 

 the directions of these causes are at right angles, is that the effects are entirely independent of 

 each other; for it will be observed that the continuous change of a cause with its exact opposite 

 necessarily introduces the idea of an impossible plane, or a plane in which the cause produces 

 no effect whatever; for it is clear that a plane equally inclined to the primitive direction and 

 the exact opposite of that direction will be a plane of indifference, or one in which tlie cause 

 produces no effect ; it may be seen therefore, without reference to the e(iuation (/?), that an oblique 

 cause may be supposed to be the resultant of two components, one in the original direction. 

 Vol. VIII. Taut III. Go 



