I 



PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 

 n = 0, p — 1, 



281 



z dfa a^ d f a 

 120 To" ~ Qz 'Ta 



When 



n=l,p=0. 



-z^d\fa cP (Pja 



480 da" 



— z dfa 



120 da 



24 da^ 



a dfa 

 60~d^ 



(fi dfa 

 6z da 



1 ^' d^fc 



a z d^fa 



a2 d-'f a 

 12 da-' 



i = 2,p = 0, 



-z^ d"- fa 



120 da^ 

 no d'^ fa a^ d'' f a 



, &c., &c. 



3 2^ 



420 da^ 120 d a^ ^ 20 da^ 

 Hence we obtain, by collecting the terms and equating their sum to \-^, 



3i^ 60 +60 da +120 da^ +i'^ + <s!^ + «S^c- 

 -^/? 



~ 3 22 



Equating coefficients of like powers of z, we obtain 



fa a dfa a^ d'-fa 

 ~ 60 + 60 rfa + 120 "Jo^ 



0. 



This equation will determine / (a), the only law of force by which a sphere 

 can attract an external particle exactly as much as if it were all collected at its 

 centre of gi-avity. 



The symbolical form of the equation is 



{D(D-l) + 2D-2}/a=0 

 or (D2 + D-2)/a=r0, or (D-1) (D + 2)/a=i0. 



Hence (D-l)/a=0, (D + 2)/a=0, 



or 



dfa . , dfa „ . 



-^^—=fa, and -^ = -'if a 



da 



da 



B 



that is/a=3A a, and/a=: — are particular integrals, and the complete integral is 

 y—h.a-v—^\ which is the law reqxiired. 



Section II. Simultaneous Equations. 



20. To effect the solution of simultaneous equations, we must eliminate one 

 of the quantities differentiated. This is best effected by treating both the differ- 



VOL. XVI. PART III. 4 B 



