450 Prof. Challis on the Principles of Hijdrodynamics, 



obtained prior to the consideration of any case of motion, it 

 ought, according to our principles, to have a general signification. 

 It may be supposed to apply to motion which is arbitrary only 

 so far as the angle 6 is determined by some arbitrary circum- 

 stance. But its application is limited by the condition, that the 

 motions obtained by giving particular values to 6 ai"e those only 

 into which the original motion defined by the equation (p) may 

 be resolved. As this last motion is dependent on no arbitrary 

 circumstance whatever, it takes place equally under every initial 

 disturbance, and any modification or resolution of it imphes the 

 operation of a subsequent disturbance. Let, if possible, 



/=acos {2 Ve{x cos 6 + y sin 6)] 



+ a'cos^ 3 v'e"(^cos- + ^ + ?/sin- +6j V. 



This is to suppose that the motion parallel to the plane of xy is 

 at each point compo\mded of two motions in directions at right 

 angles to each other. Expanding the above expression to second 

 powers of x and y, we have 



f=a. -H «' — 2oLe[x'^ co^6 + 2xy sin 6 cos O + i/ sin^^) 

 — 2a!e{x'^sv[)^d — 2xy sin 6 cos 6 + y^ cos^^. 

 By what has been said, this equality must be identical with 

 f=\-er^=\-e[x'' + y'^). 



Hence , , , ,1 



a + a' = l, and « = «' = -. 



Hence, as appears from equation [a), if o-j and a^ be the con- 

 densations corresponding to the two motions into which the 

 original motion is resolved, and S be the original condensation 

 on the axis of z, 



S — 



o"i= qCOS {2 \/e(a;cos ^4- 7/ sin 6)}^ 



'^i— 2COS {2 \^e{y cos 0— A'sin 6) } _ 



If the expansions had been carried to higher powers of x and 

 y, the two values of / would no longer have been identical. 

 Hence we may infer that the solution (tt) is applicable only to 

 points very near the axis of motion ; and that the motion which, 

 for very small values of r, is defined by the equation /= 1 — e?-^, 

 may be resolved into two sets of motions, alike in all respects, 

 but parallel to two planes at right angles to each other. 



If these results be hypothetically applied to the undulatory 

 theory of light, the original motion contiguous to the axis, and 

 symmetrically disposed about it, corresponds to ordinary light,^ 

 and the resolution of this motion corresponds to polarization. 

 [To 1>e pontinuetL] 



