202 POPULAR LECTURES AND ADDRESSES. 



parallel to that line. But if you place your eye 

 in the line of the vibration of a spherule, situated 

 about the middle of the tube, you will see no 

 light in that direction ; but keeping your eye in 

 the same position, if you look obliquely towards 

 either end of the tube, you will see light fading 

 into darkness, as you turn your eye from either 

 end towards the middle. Hence, if the exciting 



" (f) Show that every possible solution is included in . the 

 following : 



dfy d<f> d<p 



doc ' dy ' dz 



where u, v, TV, are such that 



du dv dw 



" Find differential equations for the determination of </>, u, v, w. 

 Find the respective wave -velocities for the ^-solution, and for the 

 (u, v, w)-solution. 



" (d] Prove the following to be solutions, and interpret each for 

 values of r \\J (x* + y* + z 2 )] very great in comparison with A (the 

 wave-length). 



( d(b _ d<b d<b 



\ a, = > ft = *X = > 



(.) I d * , * * 



where </> = - sin [r- t *] (k + | )]. 



(2) ] a = ' * = ~*' 7= ^ 

 J i I . 2ir 



I where \|/ = - sin \r - 1 J ri\. 



V A 



"~ I * I* sIZ> * ~ ^~.i..> y ~ dxaz 



