824 



SCIENCE 



[N. S. Vol. XLIII. No. 1119 



SPECIAL ARTICLES 



A NEW FUNDAMENTAL EQUATION IN OPTICS 



All the elementary text-books in physics 

 are still using an equation for the conjugate 

 foci of spherical lenses which is too inaccurate 

 for the calculation of microscope objectives, 

 applying with approximate accuracy only to 

 very thin lenses. The same equation is all 

 that is given in the more advanced treatises 

 except those securing a closer approximation 

 by the methods of higher mathematics. 



It is possible, however, to develop a simple 

 and rigorously accurate equation by geometry 

 applicable to lenses of any thickness by the 

 simple expedient of measuring the focal dis- 

 tances from the center of curvature of the lens 

 instead of measuring it from the surface as has 

 hitherto been the practise. 



The equation is 



f^f 



n — n cos a 

 r cos 6 ' 



in which n and n are the indices of refraction, 

 / and /' the focal distances of conjugate foci, 

 r the radius of the lens, a the focal angle and 

 h the radial angle. The meaning of these 

 terms will be further explained below. 



In the figure EQC represents the paths of 

 a ray of light refracted at the point Q on the 

 surface of a lens whose radius is OQ. 



OF and OF' are drawn parallel with QC and 

 OE respectively and in the triangle OQF 



OF ^n 

 QF n' 



since they are the sides opposite the angles 

 OQF (=180°— angle of incidence) and QOF 

 {= OQF' the angle of refraction). 



DD' is drawn through 0, making QD = QD', 

 and the angle QOD is the one designated above 

 as the radial angle. QP is perpendicular to 

 DD' and 



cos 6 = — . 



r 



Since OFD is similar to OF'D' and QF=0F', 

 OD - OD' OD 



OP 

 OD' 



OP = 



X'-3 



OD n- n' 



OD = 2 cos h -^^. 

 n — n 



EE' and CC are drawn through 0, making 

 equal angles with DD'. EOD is the angle 

 designated above as the focal angle for the 

 focus E. GH is drawn perpendicular to DD' 

 and lines are drawn from G and H parallel to 

 EQ. Since these three parallel lines are equi- 

 distant along GH, GO = OJ. The triangles 

 OJH and OOE are similar and 



qC^ _ OC~CG ^ 20C _ 

 OE OC + CG~OC + CG 



Dividing by OC and substituting OG for 

 00 + OG gives 



The 



and 



1_ ^2_ 

 0E~ OG 



1 

 OC' 



OD 



OG{=OH) 



OE^OC 



cos a 

 "OD- 



Substituting the value of OD found in the 

 last paragraph gives 



n — n' cos a 

 cos 6 ' 



From similar triangles OCD and OE'D' we 

 have 



which substituted in the above equation and 

 multiplying by n gives the form of equation 

 desired, 



OE ^ OE' 



cos a 

 cos 6' 



and 



and it only remains to be shown that E and E' 

 are conjugate foci. If the triangle EQE' is 

 rotated upon EE' as an axis, at all points on 

 the circle described by the point Q on the sur- 

 face of the lens the light radiating from one 



