368 The Magnetic Field of any Cylindrical Coil. 



including coils of circular and rectangular sections. But in 

 the case of rectangular coils the formulae become integrable. 



Let p be the perpendicular distance of any point P from 

 one of the faces of the rectangular coil, and a one of the two 

 parts into which the corresponding side of a cross-section is 

 divided by the perpendicular from P. Then the longitudinal 

 force at P is given by the algebraic sum of sixteen terms, 

 each of the form 



v . - C a dx 



\Jo (afi 



= msin 



(# 2 +j9 2 ) V^+p^ + h*' 

 ah 



And the transverse force at P is the resultant of eight terms, 

 each of the form 



J W# 2 +i> 2 +V yV+j^ + V/ ' 



2 ^{pz + hf ' a + V^+p + V/ 



The first formula in the paper can be used to find an ex- 

 pression for the force due to a circular current, at any point 

 P in the plane bounded by the circle. Draw any chord 

 through P, and call its segments r, /. Write c for the 

 distance of P from the centre, b for half the minimum chord 

 through P, and a for the radius. Then for the force at P 

 the formula reduces to 



= gs-l V« 2 -c 2 sin 2 <9 .dd. 



This can be written 



$ being the perimeter of the ellipse with a and b as semi- 

 axes, and having some value between 2wa and 4a, according 

 to the position of the point considered. 



