PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. gl3 



_'7r 1.3 ■■■ (n-1) 

 ~T 2.4.6... n 



TT 1.2 ... n 



2 (2.4 ... nf 

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



<p{s)<x ——.log C^ 

 a z' 



1 



V« 



and (p (-) ec p 



or p'/(p) « p 



We hare 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- 

 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 coeflBcient of a constant to the index 

 2" 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-i-^^ ' "^^"^(^^ is equivalent to V^T log ^ o + J-^ - 1 V a quantity which, 



when p 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 w"» power of the distance. Retaining the previous 

 notation : 



attraction =2ir I rdrf(p) — 



^ Q 



= 27ra/'7(p)rfj) 

 »y a 



