OF MAGNITUDE AND DIRECTION. 285 



, , rfG .^ ^ t> dG I . , . , 



expressed by -j-r-; now ii 9= — , — — = — a constant quantity, which can be true oi no other 

 do 7r du TT 



law satisfying the conditions /(O) = 1 and /(tt) = - 1 . If, for instance, 9 = — - , a form which satis- 



TT 



dG 



fies the conditions /(O) = 1 and /(tt) = - 1, we have 377 = 2 — 5 , and therefore the intensity of 



da tt" 



the minus affection which is measured by would increase more rapidly as the angle approached 

 the value tt. And this also shews distinctly what is meant by saying that the change of affec- 

 tion, in such causes as we have been considering, is uniform, for this is, in fact, saying that 



dG 



-— must be constant, a condition which must manifestly be satisfied when the case is one of pure 



direction, and when, therefore, there is no reason why should increase more rapidly for one value 

 of d than another. Whether there be real cases of change of affection coming under the general 

 type represented by (- 1)®, in which this condition of uniformity is not satisfied, remains to 

 be seen. 



Q 



15. The condition of uniform change of affection is satisfied by the function G = — , where 



2a 



a is some constant angle, which, in the actual case of pure direction, is a right angle. If 9 

 have this value, we have 



/(e) = cos^+(-l)-5sin^; 

 2a 2a 



and the impossible direction is given by 



=a. 



For example, let a = — , then 



4 



f(e) = cos ze + (- 1)^ sin 26, 



a formula which represent the variation of a cause which changes uniformly, and produces exactly 

 opposite effects in directions at right angles to each other. It seems not improbable that this 

 formula may be found to represent something real : may it not represent the following case .'' 

 Suppose a disturbing cause in an elastic medium which propagates simultaneously a condensed 

 wave in two opposite directions, and a rarified wave in the direction perpendicular to them ; 

 then if <p be the condensation which would exist at a given time and a given distance from 

 the centre, on the supposition of the condensing cause only acting, may not the complete expres- 

 sion for the condensation in the direction determined by the angle 6 be 



(p cos 20 + (- ])i (psin 20? 



A rough approximation to this case would be that of a tuning-fork. 



16. A more general law than that expressed by the formula f (0) = (- 1)^ is given by 



fie) = »«(- 1)® + m'(- ly + &c. 

 There is only one example of this formula which I shall notice : 

 Suppose we have a quantity P determined by the equation 



PfiQ) = acos + (- 1)4 6 sin (D), 



which comes under the above form of f(0) for the equation, may be written 



a + b ? a - b » 



