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 will equally illustrate the importance of our formula, 

 we shall 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 

 n^^ 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 ; 



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

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



U a 



Now Q^ = a^ + r^ .-. dr— — ^— ^ 



Vg'-d 

 and attraction = / / „ ^' ^ - 



-Jl c^ 



~T 7=0 



.a. ^^dgfig)f 



d.l 



dg_ 



^-dgf{g)g' 



V n2 /i2 



«" f 



Now let ^= e , ^=z, gV(g) = (P (6) 



1 r dScbie) 



therefore attraction ^-k I T7"tr^~ ' 



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



(-1)* /- sin(/w + l)7r d-^ct> {z) 

 attraction = 9 \ — • ^;: —h-^ 



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



Let it be equal to P«r» = — = P^^ 



hence (-l)Wi_ sm(m+J)7r d~:L\l^=Y^^ 



2 sin m tt ^i g— 2 



VOL. XIV. PART II. ^ 5 T 



