Horizontal Candle-Power of Incandescent Lamps. 211 



P by half the length of the filament be denoted by <f>, and let the 

 angle APC be denoted by ft. Let the intensity of the light 

 sent out or the candle-power per unit of length of the filament 

 in a direction normal to itself be denoted by I H , then the 

 illumination on the photometer-disk, assuming it to follow the 

 above mentioned cosine law, due to an element dl of the fila- 

 ment at C subtending an angle dft at P will be given by the 

 expression 



I H cos 2 ^ _ T H cos 4 /3r// 

 (UP) 2 D 2 



Also from the geometry of the figure we have 



cos 2 /3 



Hence the illumination on the photometer-disk due to the 

 element dl of the filament at C is given by the expression 



I H V.cos 2 ftdft 

 D 2 



This is the same illumination as that which would be given 

 by a vertical element of length J) dft placed at A, but having 

 a horizontal intensity or horizontal candle-power per unit of 

 length equal to I H cos 2 ft. 



To obtain the whole illumination on the photometer-disk 

 due to the whole filament of length 21, we have to integrate 

 the above expression for one element, between the limits 

 ft = and ft = (j>, and then multiply by 2, where </> is half the 

 angle subtended by the whole filament. 



Since r$ sin c/> cos </> ^ <£ 

 cos 2 ftdft= ~ -+z>> 



and since I cos <p = D sin <£, 



we see that 



f ® I U T) cos 2 ftdft _ Th 9/ /cos 2 ^ $cotc/>\ 



J . D 2 -\j* m \-T +— TV 



Hence the correcting factor when photometering a filament, 

 the apparent length 21 of which subtends a finite angle 2<£ at the 

 photometer-disk, is J(cos 2 <£ + <f) cot <£). Accordingly the ratio 

 of mean spherical to mean horizontal candle-power in this 

 last case is given by the expression 



Is_7Tf 2 \ 



1 H 4\cos 2 </> + </> cot $ J ' 



If the filament is not a straight filament but a loop as in 



P2 



