346 PROFESSOR CHALLIS, ON THE MOTION OF A SMALL SPHERE 

 and consequently by (11), 



tff = ~ £ "l+a /a'(l + 3) 2 '</* T'm ,e '■ 



Putting for shortness' sake n 2 for *•- — j, we shall have 



d*x n*r$ dx , „>. -i 



-=- + ,. • -T- + w 2 « = - rc 2 AS<? r . 



a£ 2 « (1 + a) a£ 



r* 



By integrating this equation and neglecting terms involving — , 



which will be wholly insignificant, it will be found that 



dx - " M '- 



— - = — hue 2 "( 1+s >. sin nt, 



at 



by means of which equation the decrements of the successive arcs of 

 vibration may be calculated. It is remarkable that for an incompress- 

 ible fluid, for which a is infinitely great, there is no decrement of 

 the arcs excepting so far as it arises from friction and capillary at- 

 traction. The index of e in the equation above is too small to account 

 for the observed decrements in air, which must be mainly owing to 

 friction. 



7. As another example, let the velocity = m(Z(l — e~ yt ) + m<p(t) 

 Then 



e'^(t) = ffi Wye-*' + (j>'(t)} dt = -&- e^~^' + fe r '<p'{t) dt. 



r 



Hence f (/) = EZ + e r fe r <p' (t) dt, 



a 



r ' 



a 



7 



r 



and the accelerative force of the resistance is 



ami f /3 7 v . ~£ r a T^w*\j* i " 

 I s ' e~ yt + e r J e r (p (f) dt — ke 



£-* 



If 7 be an exceedingly large quantity, in which case the sphere's 

 velocity after a very short interval is m {/3 + <p(t)}, the above result 

 becomes for all values of t which are not exceedingly small, 



