460 Prof. Challis on the Fundamental Ideas of 



which can be exactly integrated, is employed, the application of 

 its integral being restricted to large values of r, for which the 

 two equations are clearly identical. But, as is intimated in the 

 article above referred to, the result thus obtained is also deducible 

 by employing only the first equation, if the point of no velocity 

 be taken on the axis of the vibrations. The two processes are 

 therefore confirmatory of each other; and as the latter is the 

 more direct way of arriving at this important physical determi- 

 nation, I have thought proper to give it here in detail. 



At the end of an article in the Philosophical Magazine for De- 

 cember 1852, reasons are given for concluding that a set of lon- 

 gitudinal and transverse vibrations symmetrically disposed about 

 an axis may, for points contiguous to the axis, be resolved into 

 two equal sets the transverse vibrations of which are at right 

 angles to each other, and that each set may be regarded as inde- 

 pendent of the other. (This in fact is the theoretical explana- 

 tion of the polarization of light.) By reference to the same 

 article, the equations applicable to a set of vibrations the trans- 

 verse motion of which is parallel to the axis of «r, will be found 

 to be the following : — 



4> = m cos q(z—fcat + c), tc= (1 + -^J , « 2 cr-f/-^- =0, 



/- , df A<b 



/=cosVe*, u= *di' w = f Tz 



o 



q being put for -rr. Hence 

 A, 



w= — mq cos 2<s/e a? sin q{z— /cat + c), 



u= — 2m\/i sin 2\/e<z cos q{z — Kat + c) y 



aa = — mq/c cos 2\/e a? sin q(z — icat + c) . 



Suppose an exactly equal set of vibrations to be propagated in 

 the contrary direction, and let w', u', a-' be the velocities and 

 condensation resulting from the two sets. Then measuring z' 



from a point of no velocity on the axis, and substituting — r , or 



tcX! . 



q' f for 2\/e, and k' for -j- , the following system of equations 



may be formed : — 



w 1 — —2mq cos q'x sin qz' cos q/cat, 

 aa' = 2mq/c cos q'x cos qz f sin q/cat, 



u'= —2mq' cos gz f sin q'x? cos q'/c'at, 

 aa' = 2mq l K t cos qz 1 cos q'x sin q'dat. 



