Mr. J. Stuart on Galvanomagnetic Attraction. 219 

 from its centre outwards, then 



X= — - sm 0— -=- cos 0, 

 r . <x0 cm* 



cos + — - sm 0. 



r.cld dr 



To calculate these quantities, we know that 



P i= =cos 



P 2 =|(cos 3 0-|cos0\ 



-d 63/ 5/1 10 3fl . 15 a \ 



P 3 = — (cOS 5 0-— COS 3 0+ — COS0J. 



We shall only consider the case of those points for which r is 

 greater than a. Substituting these values in the expression which 

 in such instances holds for U, we have 



tt o 7 f 1 a 2 ' . 15 a 4 / 3 a 3 a \ 

 U=2ttJc {■— s - — cos0+ — • -H cos 3 0— -cos0 

 i 2 r 2 16 r 4 V 5 / 



-Sv("-.»-?—+S"») 



• ♦....}. 



Erom which, after some reduction, we obtain 



^-=-^(-l + 3cos 2 0).^+i.(9-9Ocos 2 0+lO5cos 4 0)^ 



ztt/c 2 r 3 16 r 5 



_1 (-75 + 1575 cos 2 0-4725 cos 4 0+3465 cos 6 &)- 7 

 128 v 



+ , (1) 



Y_. , . f . 3 .. n a 2 1 



27rh' 



= sin0.{ +?cos0.^-i(-27cos0+lO5cos 3 0)^ 



t 2 r 6 16 r 5 



+ -i-(525 cos 0-3150 cos 8 0+3465 cos 5 0) - 



- } (2) 



Each of these expressions consists of a series of terms in ascending 



powers of - which will be converging. 

 r 



"We shall now seek to find X and T for a galvanic current tra- 

 versing a wire coiled into the form of a hollow cylinder, of which 

 the internal radius is b, the external radius b+c, and of which the 

 length is 2/. We shall suppose the individual turns of the wire to 

 lie so close that each may be regarded as an exact circle. 



Let A B be the axis of the coil, so that A and B are the centres 

 of its two faces ; then A B=2/. Let O be the middle point of A B. 



