Telescopes and Spectroscopes for Lines of Finite Width, 329 



existence of this factor necessitates a considerable modifica- 

 tion of certain statements based on the old formula for purity. 

 Instead of diminishing continuously with increased slit-width, 

 the purity of the spectrum first actually increases up to the 

 point 



and is still equal to the theoretical resolving power of the 

 instrument when s^r = ^\*. As the slit is widened still 

 further, the purity begins to diminish, although much less 

 rapidly than is indicated by the old formula for purity. In 

 his remarks on the practical purity of a bright line-spectrum 

 in the article " Spectroscopy" (Enc. Brit. vol. xxii. p. 374), 

 Schuster says : — "The maximum illumination for any line is 

 obtained when the angular width of the slit is equal to the 

 angle subtended by one wave-length at a distance equal to the 

 collimator aperture. In that case syjr = \ and the purity is 

 half the resolving power. Hence when light is a consideration 

 we shall not as a rule realize more than half the resolving 

 power of the spectroscope." Equation (16) shows, however, 

 that under this condition for maximum illumination f the 

 purity is really 75 per cent, of the theoretical resolving power 

 instead of 50 per cent, as indicated by Schuster. A similar 

 erroneous conclusion (based upon the commonly accepted 

 formula for purity) was drawn by the writer in one of his 

 earlier papers J, in which it was stated that the purity in case 

 of stellar spectra could never exceed one-third the theoretical 

 resolving power (unless the slit-width is made less than 

 the diameter of the diffraction-image of the star). Equa- 

 tion (16) shows us that this limit should be nearly one-half 

 instead of one-third. 



Third Case. — If the radiation is not monochromatic, but is 

 made up of wave-lengths ranging over a interval from 

 X to X + AX, the dispersion of the spectroscope train will 

 spread out the image of an infinitely narrow slit into a band 

 in which the distribution of intensity (supposing the dispersion 

 over the small range AX to be strictly proportional to X) will 

 be the same as in the source of radiation. This image will 

 be further broadened by diffraction, and the distribution of 

 intensity in the image formed by the spectroscope objective 



* Unfortunately it is not generally possible to profit by this fact, 

 because for such narrow slits the spectrum is in most cases too faint to 

 be well seen. 



t As is readily seen, this condition holds only for absolutely mono- 

 chromatic sources of radiation (see ' AstrophysicalJournal,' January 1895, 

 pp. 62, 63). 



| ' Astrophysical Journal/ January 1895, pp. 68, 69. 



