Prof. Challis on the Undulatory Theory of Light. 331 



objection, it is certain that the equation obtained by Professor 

 Stokes for large values of r, viz. 



f= {2irr Ve) ~~~ *(cos 2 \/er+ sin 2 Ver), 



approaches the true integral in proportion as r is larger. In 

 fact this equation is the exact integral of the differential equation 



dr 2 rdr J r 2 



which evidently differs less from the foregoing differential equa- 

 tion as r is larger. Hence it appears that the consecutive large 

 values of r which cause /to vanish, increase by the common dif- 



ijr 



ference — — =, By peculiar reasoning applied to the infinite 

 2 ve 



roots of the equation /=0 (Phil. Mag. for February 1853), I 



found the common difference to be ultimately — E , But from 



ve 



the preceding argument it must be concluded that that reasoning 

 is not legitimate, and that some error is involved in the treat- 

 ment of the infinite roots. It will therefore be necessary to de- 

 termine the rate of propagation by a new investigation, employ- 

 ing for the purpose the above expression for/. This I proceed 

 to do by a course .of reasoning analogous to that which was fol- 

 lowed in the previous investigation. 



In the article already cited, containing Proposition X., the 

 following equations to the first approximation are obtained : — 



<f> = mcos— (z— icat + c), tc=\/l+— 5-, « 2 o-+/-^- = 

 A, V 7T at 



Also if w and co be respectively the velocities parallel and trans- 



verse to the axis of z, we have w =f S? and co = 6 -j-. The fore- 



J dz r dr 



going expression for /may be put under the form 

 (km % \/e) ~* cos ( 2\/er — t ) ', 



and as r is assumed to be very large> it may be supposed to have 

 the constant value r outside the cosine, and the general value 

 r + h under the cosine, h being always very small compared to 

 r . Then putting, for brevity, f for the constant coefficient, and 



J for 2 V*lf q — 4> we shall have 



- df - 



/=/ cos (2 Veh + d), j r = -2 Ve/ sin (2 Veh + c'). 



Z2 



