35 () Prof. F. L. 0. Wadswofth an the Effect of 



through tlie prism-train : 2nd. that the material of the latter 

 is of the same temperature and refractive index throughout. 



The general theoretical results agree so closely with those 

 obtained in practice that it is certain that the error, if any, 

 introduced by either of these assumptions must be small 

 under usual conditions. It is not, however, a priori certain 

 that this would be true when the absorption of the glass, or 

 the differential temperature-change in different parts of the 

 prisms, becomes abnormally large, as it may do in certain 

 regions of the spectrum, and under conditions of use of the 

 instrument such as are met with in astrophysical work. The 

 following investigation was undertaken in order to determine 

 the actual magnitude of the effects to be expected from large 

 variations in absorption and optical density over the transmitted 

 wave-front. 



The most general expression for the distribution in 

 intensity in the image of a point formed at the focal plane 

 of a telescope is 



I2= [jji sin r {(d ~ p) dx dy l> ■ ■ ■ (1) 



where i is the amplitude of vibration of any element dx, dy 

 in the wave-front : p is the distance of this element from a 

 point p in the focal plane at which the intensity is desired. 



In reducing this general expression to the form usually 

 given*, the following assumptions are madef : — 



1st. That the amplitude of vibration is constant, / = const. 



2nd. That the wave-front passing the diffracting aperture 

 is truly spherical, i.e., 



oc°- + ,f + f=f. 



3rd. The light which unites to form the image is strictly 

 monochromatic, i.e., \ = const. 



Under these assumptions (1) may be reduced at once to 

 the form given by Lord Rayleigh, 



where f, rj are the coordinates of any point in the focal-plane 

 image. 



* Rayleigh, article " Wave Theory," Enc. Brit. vol. xxiv. p. 430. 

 ' t See also Popular Astronomy, " Problems Relating to . . . Resolvino- 

 Power of Telescopes," vol. v. pp. 528-536 ; and Astrophysical Journaf. 

 vol. xvi. pp. 266 et seq. 



