ON THE DIRECTIVE POWER OE MAGNETS. 
495 
We shall now seek to find X and Y for a galvanic current traversing a wire coiled 
into the form of a hollow cylinder, of which the internal radius is b, the external radius 
b-\-c, and the length is 2 f We shall suppose the individual turns of the wire to lie so 
close as 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 
AB = 2/! Let O be the middle point of AB. Let P be the attracted point, PM its 
perpendicular distance p from A B. Let PAM=a, P B M=/3. 
Let C be the centre of any turn of the wire regarded as a circle of radius a , CP— r, 
PCM=0, OC=^r; then it is readily seen that for the whole cylindrical bobbin the 
forces X, Y are given by 
X P +/ p i4c 
= 1 | L dxda, 
f* J-/> 
Y C +f P 4+c 
= \ \ M dxda, 
v- J-/J* 
where L and M stand for the expressions on the right-hand side of (1) and (2) 
respectively, and where ^ depends on the strength of the current. 
To perform the integrations for the length of the bobbin in these expressions, we have 
the formulae 
p—r. sin 0, 
lx . sin$=— r . lb; 
and 
Making these substitutions for&r and r, the integrals with respect to x become integrals 
with respect to 0, which can be easily evaluated by a continued application of the method 
of integration by parts, the limits being from 0=a to 0=fo. If we then integrate the 
