436 Mr. T. Mather x>n the Shape of Movable Coils 



The ratio 



sin 6 



thus is a measure of the efficacy of the 



T 



element and its position. 



If we now draw a curve whose polar equation is 



r = x Y sin 6, 



then, as in Maxwell (vol. ii. p. 332), it may be shown that a 

 given length of wire wound within this space is more efficient 

 than if wound outside it. Hence the curve r=x 1 sin 0, 

 i. e. a circle tangential to A B at O, is the best form of the 

 section of a movable coil, just as in the case of sensitive gal- 

 vanometers the best shape is given by 



,2_ 



x 2 sin 6. 



The complete section is given in fig. 2, 

 and iconsists of two circles touching at 0. 

 The circles C and D have section-lines 

 in different directions to indicate that the 

 current passes in opposite directions in the 

 two. 



The problem may be treated in another 

 way ; for it resolves itself into finding the 

 shape and position of an area having a 

 given moment of inertia about a point in 

 its plane such that the moment of the area 

 about a coplanar line through the point is 

 a maximum. 



Taking the point as pole, and the line as the 

 ence, the expression 



of refer- 



whilst 



Nr 2 sin drdd is to be a maximum. 



r 3 dr d6 is constant. 



This may be solved by the Calculus of Variations with the 

 result above obtained. By this method Dr. Sumpner (to 

 whom the writer suggested the problem) arrived at the solu- 

 tion some time before the author considered the subject from 

 the first point of view. 



In order to illustrate the sort of improvement obtained by 

 winding coils to this shape, a table is appended, the first 

 column of which shows various shapes of section, the 

 second column the value of the ratio moment of area 

 -r- moment of inertia of area, and the fourth column the 

 deflecting moments of the various shapes all having the unit 



