jEolotropic Potential. 



453 



"With this notation 



"P \ i'k 



- -2 



77 



f//X 



•A(^) 



(126) 



expresses one of the relations (124) directly in terms of 

 j) . . . A . . . which are primary constants in the statement of 

 the analytical theorem. 



This notation is also more convenient for the transition 

 from (p to L . . . by differentiation, <£ , L having the mean- 

 ings, in (95). Thus 



-A(*)=A.AM =rA.^BT ^d2*'AW=A„^-. (127) 



Therefore from <f> = 1 

 L 



(1/JL 



, we derive 





dfi JAM _ 9 c 



J {A(^)r- dA ~ ~* a dA 



ty.) 



- L "~ VA' 



and 



>• (128), 



J 



[The same results follow of course from differentiating the 

 form in J(X), but the proof requires the elimination of an 

 integral containing X 3 in the numerator (cf. p. 437), the ultimate 

 form proceeding only to X 2 : this has been fully verified.] 



§ 34. If we set out from the equivalent form of (f> given by 



(hi 

 we have 



(124). viz.. , 1 y=-, then since .. = P, , .,=2m/, 



' 27T tlp^/w r/A 



Tims for the internal potential we have 



lp | c/&) p 1 _ A., {lx + my 4- //: )~ 



"a 





] 



• (129) 



the variables in ",,, u A being /. m, n. Now Ix+my + nz cs 

 the perpendicular t^ from the centre on a plane through 



:iijz with it- normal in the direction (Imn) : and it'.« a is the 

 perpendicular from the centre on the parallel tangent plane, 



— -A, = '\ ; hence the bracket is 1— tzr^/W 5 - Again, if o- is 



