Tlieorij of the Quadrant Electrometer. 241 



Henco, if we substitute the assumed forms for the co- 

 efficients in the system (4), we obtain 



'2.^n{an-\-h)^'^-' + ^l'2mc,,^e''—' = 0, .... (5) 

 Sr(-)"<«« + W6'"-i + 2r2mc,„6>^^— 1 = 0, . . (6) 

 t'2mchne-'"^-^ + l.'tnhne'^-^ + t^{-ynhne'^-' = 0. . (7) 



We may therefore equate the coefficients of the various 

 powers of 6 to zero. 



Equations (5) and (6) lead to the same results, namely, 



For ?i odd a„ + ^„ = 0. 



For 11 even = 2 m a^m + ^2,» + Cim = 0. 

 While equation (7) gives 



For n even =2m d2m + 2h2m = 0. 

 Hence we get for the coefficients of capacity 



Cii = «o + Sr«n^% 



022 = C^O + ^r ( — )"««^", 

 Ci3 = &0 - Sr a2n- 19"^-' + Sr b2n0'^ , 

 C2, = bo + tTa2n-lO'''-' + ^Tb2nO''\ 

 Ci2 = Co-tT{a2n + b2nW\ 



c,,=do-2trb,nO'\ 



We are now in a position to calculate the values of the 

 differential coefficients, and since for the present purpose 

 we are concerned with small displacements only, we shall 

 proceed to first powers of 6. We thus get 



^^-=a,+ -2a,e, ^' = -a, + '2bA ^ = - 2(a, + b,)e, 



|f = -«i + 2a,^, ^ = +a, + 2b,0, ^^'=-46^ 



Substituting in (2) we obtain for the force on the needle 



«i(V,-V,) {\,-i(Y, + Y,)}+e{a,(Y,-Y,y-2b,iY,-Y,)iY,-Y,} } 



Hence, instead of the ordinary equation 



F^ = ai(V,-V0{V3-i(Vi + V,)}, 

 we obtain 



{F + 2h,iY,-Y,){Y,-Y,)-a,iY,-Y,)^}0 



Phil. -Mag. S. 6. Vol. 6. Xo. 32. Aug. 1903. R 



