354 



PROCEEDINGS OP THE AMERICAN ACADEMY 



We have heretofore supposed the coil to have been made of a single 

 turn of wire. If the coil consists of a number of turns of wire, the 

 potential at any point may be obtained by 

 the following approximation. Let the single 

 turn, the potential due to which at any point 

 we found could be expressed by equation 

 [XIV.] or [XV.], occupy the centre of the 

 coil whose rectangular dimensions are | and 

 7. Let the co-ordinates of the wire at the 

 centre of the coil be x and y. Now the 

 potential at due to the coils whose cross- 

 sectional area is |rj will be a function of x and y. If we look at 

 equation [XIV.], 



Fig. 5. 



Q = 



— 27r+ ZTT cosa — 2tt ^-^ — - rP, {&) -A- &c. 



' a a cos a i ^ ' ' 



or, as it may be written, 



Q = _ 27r+ 27r^, — 27r^jrP,(d)4- &C.+ 27rQ,r'P,(^), [XVL B.] 



we see that ^g, ^j, Qi^ &c. are sxach functions of x and y. 



Let G be the mean value of Q for the values of Q for each wire 

 within the limits -\-\^i — \^, -\-\ni ^.nd — \r], or 



G 



I Qdx dy 



-h-n 



[XVIL] 



This expression gives the value of a new coefficient G for the coil 

 whose cross-section is I?; in place of Q, the coefficient for a single turn. 

 From this we see that the potential at a point due to the coil 1 17 is 



Q = — 27r-\-2^G, — 2TrG,rP,{6)-]-&c.+ 27rGir'Pi{6); [XVIIL] 



or, calling G' = 2irG, we have 



Q = — 2n-]-GJ—G,'7- P, (6) + &c. + G/ r'Pi {6) . [XIX.] 



