
PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 221 


Ped Ue tl ee Gi ee 
a ilgeaenen ete 
et 2 a fa a2. d fa psdrarian 
IP Song ie ait eae 12 a? 
a aM sf d fa ata’ fa- a d* fa 
n=2,2=0, 799 aa * 120 d@ +20 dat? 


&e., de. 

Hence we obtain, by collecting the terms and equating their sum to ae 

Chie apa ae Afia a a? fa - 
32 60. 60 da, 420 dae * + *tRAt Se. 
Equating coefficients of like powers of z, we obtain 
S@. G@4f a @ Gd fa 
0 + 00 da ti da ~? 
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 gravity. 
The symbolical form of the equation is 
{D (D—1)+2D—2}fa=0 
or (D? + D—2) fa=0, or (D—1) (D+ 2) fa=0. 
Hence (D—1) fa=0, (D+ 2) fa=0, 
or <f4_ Fa, and “14 =—2f¢ 
that is fa=Aa, and fa= 2 are particular integrals, and the complete integral is 
y=Aato which is the law required. 
SEecTION If. 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- 
VOU. Vi. PART III. 4B 
