454 Prof. Challis on the Diversion of Light. 



determination of the motion and propagation of plane waves, 

 when deduced from only two fundamental equations, presents 

 contradictory results which disappear when the third equation is 

 also employed, it is not allowable to dispense with the use of this 

 equation in any case. That one instance suffices to prove that 

 inferences from the two equations alone, even if not self-contra- 

 dictory, are not necessarily true. For the purpose of exhibiting 

 clearly the distinction between the two processes, and giving the 

 opportunity of judging of their respective merits, I shall attempt 

 the solution of the proposed problem, first, by employing only 

 the two usual fundamental equations, and then by joining with 

 them the third. 



The reasoning being restricted to the first powers of the velo- 

 city and condensation, we have the usual equations 



a*d<r du_~ a*dcr dv_^^ a*da dw ^ 



dx dt ' dy dt dz dt 



and 



dcr du dv dw _,, 



dt dec dy dz 



Differentiating the last equation with respect to t, and substi- 

 tuting from the other three, we obtain 



di* fl " W rfy 8 dz*)' 



Now, from the conditions of the problem, the condensation at any- 

 given point must be a periodic function of the time, the period 

 being the same as that of the incident waves. Hence we may 

 assume generally that v=f x (£)<£j(#, y, z) 4-/2 (0 $2 fc V> z ) + & c -> 

 provided the functions^, f%, &c. satisfy the equations 



/,"« +nVM =0, f»{t) +n%(t) =0, &c., 



n being put for — — . Since from this value of cr, -YY + n q cr=0, 

 it follows by substitution in the foregoing equation that 



doe* dy* dz* + ~tf 



As by hypothesis the condensation is symmetrically disposed about 

 an axis drawn through the centre of the sphere in the direction 

 of the incidence of the waves, its value at any point at a given 

 instant is a function of the straight line (r) joining the point and 

 the centre of the sphere, and the angle (6) which this line makes 

 with the axis. Hence the last equation may be transformed into 

 one containing only the polar coordinates r and 6. The trans- 



