238 



Mb HOPKINS ON AERIAL VIBRATIONS 



and for ar, writing ar — x, 



/{(aT + 2/)-^}=/(aT-ar)-x//{«T-(a; + c)}+0(«T + 2/-;r) (3). 



Also putting ar + ^sl—x, for ar, 

 /■{(aT + 4/)-ar}=/(aT + 2/-ar)-v|/{«T + 2/-(a; + c)} +0(«t + 4/-^) 

 =f{aT-x)-^{aT-{x-^c)} 



-x//{aT + 2/-(a; + c)} 

 + 0(aT+2/-a;) 



+ (i>{ar + 4!l-x). 

 And similarly, we have 



2/^ 



/{»(.+ ^v^* 



^{«('^ + — )-(*' + c)}, 



■■/{aT-x)-< 



f {"(-r +—)-(« + c)}, 



&c. 



, ( r 2{n-l).l-\ . 



&c. 



.^{«('^ + -|-)-^} 



