480 PROCEEDINGS OF THE AMERICAN ACADEMY 



this does not, however, give the minimum of dispersion for which the 

 condition (found by differentiating) is 



» 2 sin (a -(- r{) cos (a -j- 2 ?•,) -\- sin r r = (5) 



an equation which has, I believe, never been solved. When a is very 

 small 



sin (a -[- r x ) = a -f- rj, cos (a -J- 2 r^) = 1 sin r 2 = rj 

 since i\ is small also, and (5) becomes 



n 2 (a -4- r : ) -4- ?\ = ; r, = a 



n 2 



n 2 + 1' 



I find that this equation gives very nearly the true value of ?\, even 

 when a is large, thus for n = 1.5, a = 15°, 30°, 45°, and 60° it gives 

 r x = 10° 23', 20° 46', 31° 9', and 41° 32', while the true values ob- 

 tained by trial in (5) were 10° 23', 20° 45', 31° 6' and 41° 29'. 



To show the comparative deviation and dispersion, while a prism is 

 rotated, I have calculated the following table for prisms whose index 

 of refraction is 1.5, and their angle 15°, 30°, 45°, and 60° ; r 2 is the 

 angle at which the light leaves the prism, and i x that at which it enters. 

 The columns headed " Reduced Deviation " are calculated by the con- 

 dition that for small angles at which the dispersion and deviation are 

 proportional their units shall be the same. 



From this table it follows that if we turn the prism of a spectroscope 

 20° from minimum deviation, the dispersion will be nearly doubled, 

 thus doubling the power of the instrument. In projecting the spec- 

 trum on the screen, this device is often useful ; I have thus, by a single 

 60° prism, projected a spectrum of such a size that the two parts of the 

 D line were about a millimetre and a half apart, each having a thick- 

 ness of only about half a millimetre. 



Since the dispersion doas not depend strictly upon the deviation, and 

 we may have two prisms producing the same dispersion but unequal 

 deviations, we can evidently make one achromatize the other, even if 

 they are made of the same kind of glass, and have the same angle. 

 Thus if two 15° prisms be so placed that r 2 shall be 50° for one, and 

 37° 21' for the other, the dispersions of each will be .418, while the de- 

 viation will be 11° 7' for one, and 17° 53' for the other, as may be seen 

 from the table. Again, if the ray of light passes through first one and 



