Q12 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



» 



and, 



(-l)i ^ _ sin(m + l)'7r . ..j, rf^ z^ 

 ^ ^ 2 sinrnTT ^^^ ^^i 



In-, . n 



V+2. 



I . n— i-l 



Sin —IT— "TT 



n— 1 



(p (s) a« 



... (J-^ =J_ 



or ?7(?)=-^ 



Cor. 1. If n=l ,/(g) = — ; the law of nature. 



Cor. 2. If n=0 , f(Q)= — , or the law of force, which must hold, in order 



that an indefinite bar may produce the same effect on points at all distances, is 



that of the reciprocal of the distance. 



«— 1 fi 2 



Cor. 3. If 0(0)=z 2 . we have, by writing — ^ for m, 



, sin^TT '±± , /- +1 sin^ „_i 



(-.1)* ^-^^ ■ ^ ' =(-!)' p • 4fi ^-^-"^ 



■^ Sm — jr— TT / W , 1_ sin — ;r— TT 



^ ' 2 ''" 2 "^ 



V TT ' 2" + 2 



or P= 



/f 



1 



fn T 



If w be odd, this gives 



n — 1 » — 3 M — n — 2 



V'''" 2 ' 2 ■" 2 



/r 



p= 



?i M — 2 n — » — I 



a 



22 22 



2 (y^-l)(OT-3) ... 2 a/tt 



2 ■ w (n—2) ... 1 ■ -v/tt 

 1.2.3 ... n 



(1.3.5 .. 

 ' 7i be even, we obtain 



n — 1 w — 3 



n — n — 1 jl 



p VTT 2 2 



2 ' " " — ^ 

 2 ' 2 



••• 2 '2 



... - 1^ 

 2 ' 



