Mr. GOODWIN, ON THE PURE SCIENCE OF MAGNITUDE AND DIRECTION. 279 



Now, it may be assumed, that the effect of the oblique cause can be measured by a symbol 

 of the form Pf(9), where f{9) is a modifying function, which would be 1 if 9 were zero 

 and whose general form must be determined. This assumption appears admissible, because if 

 there be a symbolical expression for the effect of an oblique cause, it can be of no form more 

 gen>.ral than that assumed, and if there be no such symbol, this will appear by the impossibility 

 of determining the form of f(9). 



4. To determine the form of the function /, I observe, that the fundamental law of such 

 causes as we are considering is, that the exact reverse of any cause whose magnitude and direc- 

 tion are given is one of equal magnitude and exactly opposite direction ; so that, if we denote 

 opposite affections by + and - , then + P must change into — P, while 9 increases from to tf : 

 moreover, the change of P, |or rather of Pf(9)}, as 9 varies continuously, must manifestly be 

 continuous, and not only continuous but uniform, that is, the rate at which the affection changes 

 from + to — must be the same at all stages of the change, since there is no reason why the 

 change should be more rapid for one value of 9 than for another : — speaking symbolically, it 



may be said that ' ' must be the same for all values of if a be a given quantity. 



f(.9) 



This law, to which f{9) is subject, and which flows at once from the pure idea of direction, 



is sufficient to determine the form of /. For suppose the angle tt to be divided into n equal 



parts ; then, if the direction vary through one of them, the symbol representing the effect of 



the oblique cause will be P/l — j; if it vary through two such divisions, the symbol becomes 

 Pfi.-\f^-], or p|/(-j i, fit also becomes P/| — IJ, and so on; and when the direction has 



varied through n angles each equal to — , the symbol becomes ^yl ~ ) > , but by what has just 



been said this symbol must represent the exact reverse of + P, and must therefore be = - P ; 

 hence we have 



/■ -) = (— 1)" = cos- + (- nisin- , 



and if we put — = 9, 



n 



fi9) = (- l)" = cos0+(- l)*sin(9 (J). 



It will be observed, that n may be made as large as we please, and therefore, the condition will 

 1)6 satisfied of 9 varying continuously from to tt ; also the change of/ is not only continuous 



but uniform, for it h clear that the expression (- 1)" satisfies the condition that "— : — — ^-^ 



shall be the same for all values of 9 ; and this follows necessarily from the mode of determining 

 f, without assuming that /(-tt) = - l, for the fundamental law of variation of/ is expressed by the 

 ('ijuaticm 



!/(!))"=/<«■ 



