212 Dr. Fleming : Ratio between Mean Spherical and 



an actual lamp, which may be so coiled that the different 

 portions lie in different planes, we have an additional small 

 correction to make. We may first notice the effect of the 

 inclination of a straight filament in various directions. 



Let three coordinate planes he drawn through the centre 

 of the sphere of reference. Let one of these planes be 

 horizontal. Let the photometer-disk be placed with its plane 

 vertical. Let two other planes be described through the centre 

 of the sphere which are respectively parallel and perpendicular 

 to the photometer-disk plane. 



Then consider a straight filament lying in the plane, per- 

 pendicular to the photometer-disk and having an inclination 

 to it. 



It is clear that if the cosine law of radiation holds good, 

 the light sent out by each element of the inclined filament in 

 a horizontal direction will be the same as that of its projection 

 on a plane parallel to the photometric disk. Hence we may 

 substitute for the inclined filament its projection on a vertical 

 plane, and if we assume the filament and its projection have 

 the same candle-power per unit of length, the horizontal 

 illuminating power w T ill be the same in the two cases. If, 

 therefore, the filament lies in any plane we may consider 

 it divided into equal elements of length, each of which 

 makes its own contribution to the mean spherical and the 

 mean horizontal candle-power. Disregarding for the moment 

 the effect of distance from the horizontal plane on the illu- 

 minating effect of each element on the photometer-disk, we 

 see that the ratio of the mean spherical to the mean horizontal 

 candle-power must be greater for each element of length of 

 the filament the more it is inclined to the vertical plane or the 

 less its inclination to the line joining it to the photometer- 

 disk. 



Hence if the whole filament lies in one plane placed per- 

 pendicularly to the photometer axis the ratio Is/Ih must be 

 ir J 4c. If it does not lie wholly in that plane the ratio will be 

 somewhat greater. 



In the case of filaments coiled in various ways a small 

 correcting factor is also necessary, providing for the reduced 

 illuminating effect of each equal element of the filament, into 

 which the filament may be considered to be divided, the 

 further the element is removed from the axis of the photo- 

 meter, that is, from the horizontal plane. In the case of 

 simple horseshoe-shaped filaments, so placed that the whole 

 filament is visible, this correcting factor will be the factor 



2 



COS 2 (£ + </> COt (j) 



