PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



609 



-^ (l6a' R^-4r «'R' + 16 a' R^ 

 4a^ [ 3 



QO -I a oo 



-':y a^ R^ + =^ «^ R' -32 a' R' + 16 « R^-':^ « R* 

 o o o 



+ ^R'^ + 16a^R^- ^aR^+ 16aR='-^R' 

 o 5 5 



o 15 



} 



^ / 16 ,^ 32 48 ,„ 32\ ^, 

 / 48 16 64\ ^A 



''' 16 3T,3 



4 a' ■ 3 



4 

 = — TT R' . a 

 o 



= mass multiplied by distance of centre of gravity of the sphere 

 ii'om the point. 



This result is obviously correct. 



It will be remarked, that all we have effected by means of our process, is the 

 transformation of a definite integral into an indefinite one. This transformation 

 is, however, of the utmost importance as a general fact, although we make little 

 of it in the present instance. 



Next, let the force of attraction be that of the inverse square of the distance, 



then shall we have to find the integrals of , .^ ■ 



^ , 1 1 



Now, d-^ 



d- 



1 



and 



d 

 d-^ 



1 / \ r^ +a — a 



= -zlogiz + a)+J^-^-d. 



= — (z + a)\og (z + a) + z , 



