362 Prof. Challis on the Motion of a small Sphere 



not to difference of velocity, but to difference of condensation, 

 the term containing it must involve the condensation. If, 

 therefore, V = a / S=wsing r (fl^4-C), and if H and K be substi- 

 tuted for numerical coefficients the values of which are known if 

 A be given, we have finally 



The acceleration of the sphere has thus been determined, so 

 far as it depends on the terms in the original values of a and 

 w which contain the first power of m ; and it will be seen that 

 the above expression for it is wholly periodic. From this first 

 approximation we might next proceed to include terms involving 

 m 2 . But since these terms are of very small magnitude com- 

 pared to those which have been considered, we may dispense 

 with going through the details of the second approximation by 

 making use of a general analytical formula, according to which 

 if /(Q) be a first approximation to an unknown function of any 

 quantity Q, the second approximation is/(Q) +/ , (Q) AQ, if AQ 

 be very small. By applying this formula to each term of the 



d?x 

 above expression for -j-^-, we have to the second approximation 



very nearly 



ay !: , 

 dt 



= h -(? + t) +] w- ; 'H s+as > 



Now, from what was proved relative to the component vibrations, 



AV is wholly periodic, and so, by consequence, is — '-=- — ; but 



it was shown that AS contains terms that are always positive. 

 These terms consequently indicate that the sphere has a 'perma- 

 nent motion of translation. This motion is towards or from the 

 origin of the undulations according as h is greater or less than 

 unity. 



Again, it was proved that different sets of such positive terms, 

 corresponding to different origins of disturbance, may coexist ; 

 from which it follows that simultaneous undulations from differ- 

 ent sources may independently produce motions of translation of 

 the sphere. 



It also appeared that when the axes of the component vibra- 

 tions are exactly or very nearly parallel, the sum of the conden- 

 sations expressed by these positive terms is proportional to the 

 number of the axes that pass through a given area. Hence, if 

 the axes diverge from a centre, this sum, and, by consequence, 

 the corresponding accelerative action of the undulations, varies 

 inversely as the square of the distance from the centre. 



