CONTEMPORARY ADVANCES IN PHYSICS 317 



and to second approximation, 



r ^ ro - (yrj + zO/ro- (89) 



Into equation (77) we insert the first-approximation values of r and 

 cos 6 in the multipHers of the sine-function, but the second-approxima- 

 tion value of r into the argument of the sine. Therefore we have, for 

 the value of 5 at the very distant point (x, y, z), the expression: 



J fyi I X \ f f* 



^ ^ ~4iV\^ '^7 I I I ^^^^ ^^" ^^^ ~ ^^'^ ~ "^^^ "*" 2iVro). (91) 



It is expedient to introduce the three direction cosines of the line 

 extending from the origin to the field-point, the cosines of the angles 

 between it and the coordinate axes: 



a = cos (x, To) = = x/ro; = y/ro] y = z/tq. (92) 



Then, with a slight additional transformation, we convert equation 

 (89) into: 



5 = const. (1 + a) sin {nt — 'mro) I | dr]d^ cos m(^r] -f 7^) 



— cos (nt — mro) I I drjd^ sin m(^r] + 7^") 



= const. (1 -f a)[C sin (nt — mro) — 5 cos (nt — mro)], 



the symbols C and 5 being traditional for these integrals. 



The coordinates of the field-point have disappeared, leaving only the 

 cosines which define its direction as seen from the origin. This means 

 that we have here the formula for the wave-motion over any plane 

 parallel to the screen and infinitely far away, in terms of the directions 

 in which its various points are seen. The words "infinitely far away" 

 sound formidable ; but it is not necessary to depart for infinity, in order 

 to find a plane where (93) describes the state of affairs. There is an 

 artifice for bringing the infinitely distant plane up to a convenient 

 nearness ; an artifice known as a lens. When a converging lens is set 

 up before the apertures, the wave-motion predicted by the formula 

 (93) for all points infinitely far away upon the line with direction 

 cosines (a, 13, 7) — this wave-motion occurs at the point where the line 

 intersects the focal plane of the lens. Therefore we may regard equa- 

 tion (93) as the description, according to the wave-theory of light, of 

 the distribution-of-amplitude in the focal plane of the lens. (To con- 

 vert the cosines into coordinates in that plane, it is sufficient to multiply 

 each by the focal length of the lens.) 



