670 Mr. J. Rose-Innes on Integration of tlie Differential 

 Earnshaw's trial solution may also be written 



-F® 



if for shortness we denote the term -^ by"w. 

 Differentiate with regard to p, 



Then since 



(I)'- ws? 



Hence cZw 1 /d» 



dp pV dp 



du du dp 



dt dp dt^ 



we have dp _du _^_ du 



dt dt ' dp 



1 dp . du 



p dx dp 



_l_dp_dp 1 /dp 



p dp dx ' p\/ dp 



/dp dp 

 V dp 'dx' 



P_ 



PoV dp 



Now 7 dp _ dp ,. 



dp=-fdx+ -felt 

 dx dt 



-%{*■-%-/%■*}■ 



Thus dp =0 if <£b and <# are so related that rf#= —a / -^- eft. 



. p /dp . P*\ ^° 



The velocity — a / -y- occurring in the last equation will- 

 be that of a plane moving as the wave moves so as to occupy 

 successively the undisplaced position of those particles which 



