= R 



1883.] Lord Rayleigh, On the mean radius of coils. 32."> 



may be denoted by p, and is to be regarded as approximately 

 constant. 



The introduction of the factor p makes but little difference in 

 the investigation of § 700. If we take the origin of co-ordinates 

 x and y, no longer at the geometrical centre, but at what may be 

 called the centre of density of the section, we shall have (as in the 

 ordinary theory of the centre of gravity) 



jfpocdzdy = 0, ffpydscdy = 0, 



the integrations beimr extended over the area of the section. If P 

 be any function of x and y, P the mean value of the function (with 

 reference to p), P the value at the origin, we have 



PJJpdxdy =JJPpdxdy 

 ° jjpdxdy + % -^- % jjpafdxdy + ^— J J pxydxdy 



+i di*j! pf<ixdy ' 



the terms of the first order disappearing in consequence of the 

 choice of origiu. In the terms of the second order we may neglect 

 the effect of variable density, and write 



ffpx'dxdy = j^ PJJpdxdy, 



Jjpy 2 dxdy = ^ rfjfpdxdy, 



fj pxydxdy = 0, 



|, t] being the breadths in the directions of x and y of the rect- 

 angular section. Thus 



d 2 P , d 2 P^ 



dx Q ' + V dyl 



The form of this expression is the same as when the windings 

 are supposed to be distributed with absolute uniformity, but the 

 mean radius and mean plane are to be reckoned with reference to 

 the density of the windings. 



In the application to the galvanometer-constant of a coil, we 

 have, if A be the mean radius, £ the radial and 77 the axial 

 dimension of the section, 



P=P + lh(?^+V' 



^-"AV + ^A n 8 A* t 



by means of which, f and 77 being approximately known, G\ may 

 be inferred from A, or conversely A may be inferred from Q y If 

 the ratio of galvanometer-constants of two coils has been de- 

 termined by the electrical process, the ratio of mean radii can be 

 accurately deduced by use of the above formula. 



