/Eolotropic Potential. 457 



This makes 



~dyjr . (/+,. koX* 1*1(1* + my + nz)dm , 1q(n 



- T =0, and gives ; = — ■: — I— — - — — ., . (lay; 



^/Z ° rftf 4W. J Up(u A + flUp)% 



The parr of V/*^ due to explicit differentiation with regard 

 to ,<//:. V .'• ] yfr being 



/ rf r d , d\ilyjr ( , d \dyjr . 



"^J(« A +^ (AW)' v/ J W 



The part due to implicit differentiation through jjl is 

 U P \, Cd^lx + my + nz) | / dj. dp d^ ^ 



which by (138) is 



A-p r , f t n . .( dn ,d[M , dp\ 



spfeL^+^+^TO + '' * + ' y </t) + • • -J 



Thus V 2 ^ is zero when X or //, is variable ; when X is con- 

 stant only the first part exists, and this for X = is reduced 



to —kp. The vanishing value of ^- when (138) is used has 

 the form ° X 



I'P A a kp l-u a 



r(±— "a) or 



4 [A(X)]i l "' 4 ^(Xj 



agreeing with the original solution. 



The formula is o£ sufficient interest to justify a separate 

 statement of the isotropic case for elementary work. Taking 

 the usual notation as to density, we have to show that 



. p,fi ! 'C i dco r (/.*■ + w // + ;/ .: )--[ 



*" 2 J(2a 2 P+X)iL ^r/-iX "J 



satisfies the differential equation V -yfr = 0, when X is a variable 



a? 

 given by 2— 5 , . =1; and that if X is omitted from the 



Phil Mag. S. 6. Vol. 9. No. 52, Jy,/7/ 1905. 2 H 



