346 Prof. Challis on the Motion of a small Sphere 



By substituting the foregoing value of ty, I found, by calcula- 

 tions somewhat intricate, that this equation is satisfied to terms 

 inclusive of m 3 , iff, g, and h have values expressed by the fol- 

 lowing series : — 



f^l-er 2 



eV f m 2 / 5-k 2 9/e 2 -29 \ \ 

 + 4 I. a 2 \6(^-l) 2+ 24 )} 



e 3 r 6 T m*( Y7-\\& 147^-247X1 

 "" .36 L fl ? \3(« a -l) 2+ 24 ;/+ &c - 



g = 1 - er 2_ 3>fp± eh A + 12 f + 1 6 3 r 6 + &Cv 



4 1© 



(It should here be noticed that the value of /,, or g, obtained 

 in the January Number is incorrect.) These equations prove 

 that for points near the axis it is legitimate to suppose, as I have 

 done in previous investigation s_, that yjr^fcf), cf) being a function 

 of z and /. As the foregoing reasoning may be carried on 

 ad libitum, we may conclude that for this kind of motion 

 udx-\-vdy + wdz is a complete differential for the exact values of 

 u, v, and w, and that the motion, being characterized by this 

 analytical condition, is independent of arbitrary disturbances. 

 Since it has been shown in the Number of the Philosophical 

 Magazine for May 1865 that the constant k is determined by 

 the equation k 6 — /c 4 =1, it is easy to convert the foregoing factors 

 that are functions of k into numerical quantities. 



By means of this determination of the function ty, and the 

 equation • 



the values of the velocities w and g> respectively parallel and 

 perpendicular to the axis, and the condensation a, may be found 

 for a point at any distance from the axis. For our present pur- 

 pose it will suffice to regard m as extremely small compared to 

 a, and to restrict the calculation to terms involving m 2 . Accord- 

 ingly the velocity and condensation at any position and at any 

 time are given by the following equations : — 



g± i ■■„ #*■_, §^ri e v 4 12 f u + ] «v» + &C, 



4 lo 



