318 



Prof. Challis on the Motion of a small Sphere 



dr 2 rdr 



+4*1/1=0, 



dr* 



+ S- +i( v.- 



l d ll- 



2k 



A« \ dr 2 



/ft, 4 ), 



a/d 2 Q 



KXdtf 2 





+ dz* a* dt 2 ) ~ 



2. 





+ sin 



Here/j, e„ ^j, and g, are the values off, e } g, and # for any one 

 value of X; and in the last equation q v £,/,, and g 2 , £ 2 , / 2 are 

 the values of q, £, and / for any two values of X. The first two 

 equations prove that the series for/^ and g x are respectively of 

 the same form as the series above forf and g. The equation 

 for determining Q may evidently be satisfied by assuming that 



Q = 2 . [R sin (ftfi+tfaft) + S sin (ft &-&&)]> 



R and S being functions of r, the values of which may be ob- 

 tained by the method of indeterminate coefficients, every such 

 function as/ t and/ 2 having been previously determined. 



Having thus found for ty an expression which is applicable to 

 composite vibrations relative to an axis common to all the com- 

 ponents, we may proceed to deduce expressions for the compo- 

 site longitudinal and transverse vibrations (w 1 and co ! ), and the 

 composite condensation (a'). By means of the equation (B) the 

 following results will be obtained without difficulty : — 



w 



' = m£ . [/sin gf] 



2w?-V v r oi n. 2 ^Q 



^--■ffi-^^tt^]^ 



qdr 



rl — 



^^■u^^-^-[i^]y- 



In these equations k 4 has been put for 



1 



-1 



the equality of 



the two quantities having been proved by the equation k 6 —k 4 = ] . 

 Now, since 



(2 . [/sin ? ?]) 2 =2 . [/WffQ +22 . [/./.riny,?, sin,y ,and 



(H£^]h 



s -[^£ cos5 



Kf + . s ?-ISS«»ft.'6 , "*6] 



