PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 613 



_^ 1.3 ■■■ (n-l) 

 ~ 2 2.4.6... n 



^ TT 1.2 ... n 



~'2 ■ (2.4 ... nj 



1 c 



Cor. 4. If the force of attraction a log — : let it equal log — ; 



a' 





.. ^^.;... ^^^ 





1 



a 



• • 



'^CO'^' 



and 



^(pO"^ 



or 



?^/(?) ^ ? 





A9)^j' 



di 



logC^ 



We have written down this case because at the first sight it appears anoma- 

 lous. We know indeed that, when the force of attraction is constant, log . — en- 



a 



ters into the expression for the whole force ; but we must remember that this 

 force is infinite, so that it does not in reality vary as log . — . But, in addition 

 to this, we found above that, when the force of attraction varies inversely as the 

 distance, the attraction on a point is constant. But the anomaly is easily ex- 

 plained when we reflect that the differential coefficient of a constant to the index 

 — is of the same form as that of the logarithm of z ; and further, that the actual 

 value of the attraction is expressed in the form of a circular function, viz. 

 /^cos-^— ) , which is equivalent to V^-l log (a + J—^ - 1 j , a quantity which, 



when ^ is 00 , varies as log a. 



Let us now pass on to the more ordinary problem of determining the law of 



attraction, by which the whole attraction of an infinite plane on a point without 



it, may vary inversely as the n^^ power of the distance. Retaining the previous 



notation : 



attraction = 2 tt / rdrf{g) — 



u q 



= 27r/ af{g)dQ 



'J a 



= 2'rraf^f{§)dQ 



