PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. gj] 



We cannot expect, in our present state of knowledge of the subject, to de- 

 termine any converse propositions in so general a case as that of the sphere ; but 

 as a more simple example wiU equally illustrate the importance of our formula, 

 we shaU give one. 



A homogeneous rod of small thickness attracts a point without itself : it is 

 required to find the law of attraction, so that the whole force may vary as the 

 w"i power of the reciprocal of the distance of the rod from the point. 



Let a be the distance of the point from the nearest point in the rod ; 

 r, r + d r, co-ordinates of two points in the rod ; 



g the distance of the point whose ordinate is r from the attracted point : 

 Then, if/(p) be the law of attraction, we obtain whole attraction 



= r^rfig)± 

 U a <^ 



Now p2 = a2 + »-2 .-. dr = -fi££=^ 



and attraction = / - /{ °' — 



Ja Vg^—a^ 



1 



2 J a Vf 



d.\ 



^1 f ^^:^?{^ 



2 7» n — r 



V «2 n2' 



Now let ^= e , ^=., QV(g)=ct> (6) 



1 r- d6d)(6) 

 therefore attraction —-k J 77=33-"" " 



Hence this form coincides with that in our theorem, and we get 



o^ 



attraction =^ H ^^^ ^^^ 



But, according to hypothesis, the attraction must vary as a~". 



P -1 



Let it be equal to Pa-" = — = P^^ 



hence (-1)4 /i_ sin>H4)j7r ^J^i'^^ = p s'? 



2 sin m it d j— ? 



VOL. XIV. PART II. 5 T 



