Nov. 30, 1871J 



NATURE 



87 



From these piinciples it is easy to deduce a formula which will 

 express the dispersive efficiency of a given instrument, and 

 enable us to judge of the effect of variations in the proportion 

 and arrangement of the parts. 



Let f be the focal length of the collimator. 

 /' ,, .• ,1 telescope. 



m the magnifying power of the eye piece (which is found 

 liy dividing the limit of distinct vision by the equivalent 

 focal length of the eye-piece and adding unity to the 

 quotient). 

 n the number of prisms in the train. 

 Tc the widtli of the slit. 

 1; the luiiiitnutn visibile above alluded to. 

 d (I, the difference between the indices of refraction for 



t«o adjacent lines ; and finally 

 5, the co-efficient of dispersion for each prism (which, r 

 being the refracting angle of the prism, is given by the 

 equation 



.,.) 



\/ I -jn^sin- 



If, now, we put D for the distance between the centres of the 

 two lines, and /' for their breadth, we shall have 

 D = m » s 5 /'. d ft, and 



But the distance between the edges of the lines equals D- b ; 

 and this, for two lines as close as the instrument will divide, 

 must equal /•. 



Hence /■ = in n S/^. d fi - " - 



/ 



Finding from this the 



(I) 



d E = »f , H. S 



(^) 



value of d /i, taking its reciprocal as a measure of the dispersive 

 efficiency of the instrument, and calling it E, we get 



E = ;« nh — -fl : 



- kj + mwj 



This formula, in which jii, n, and S appear as simple factors, 

 of course supposes that the perfection of workmanship and 

 intensity of the light are such that there is no limit to the 

 magnifying power and number of prisms wliich may be em- 

 ployed. 



My special object, however, in working it out has been to 

 exhilJit clearly what is evident from its last term, the dependence 

 of the dispersive efficiency upon the focal lengths of collimator 

 and telescope. 



Differentiating equation (i) with respect to/and y"', we obtain 



(kf+mw/^y- I 



which shows that any increase in either y'or /"' adds to the dis- 

 persion. If yincreases, both D and !> increase in the same pro- 

 portion, and so, of course, does the width of the interval between 

 the adjacent lines; while every augmentation of y ' decreases the 

 wiJih of the spectral images without in the least affecting the 

 distance between their centres. 



This princip'e seems to hive been often overlooked, and colli- 

 m.^ll)rs and telescopes of short focus employed when longer ones 

 would have been far better. 



Ill spictroscopes designed to he used for astronomical purposes, 

 at ilie principal focus of a telescope, there is, of course, no 

 advantage in making the angle of aperture of the collimator 

 nuich greater than that of the equatorial iiself; accordingly a 

 collimator of one inch aperture ought to have a focal length of 

 10 or 12 inches, or, if special reasons determine a focal length of 

 only 6 inches, then it is needless to make the collimator and 

 view telescope much over half an inch in diameter, and the 

 prisms may be correspondingly small. 



If, on the other hand, the focus of telescope or collimator is 

 lenglhened for the purpose of securing increased dispersion, 

 object glasses and prisms must also be correspondingly enlarged, 

 in order to transmit the same amount of light. 



It is, perhaps, worth noting that when y' and y'l are equal, 

 formula (i) becomes simply 



E = 



5./ 



k + mw 



(3) 



Luminous Effidency. — The extreme faintness of many spectra 

 greatly embarrasses their study, so that it becomes a matter of 

 interest to examine how the different dimensions and proportions 

 of a given instrument stand related to the brightness of the 

 spectrum produced. 



It appears to be_necessary, for this purpose, to distinguish two 



classes of spectra, those composed of narrow and well defined 

 bright lines, and those which are not, the light being spread out 

 more or less evenly and continuously. 



The brightness of a spectrum of the latter kind is evidently 

 direcUy proportional to the amount of light admitted, diminished 

 by its subsequent losses, and inversely to the area over which it 

 is distributed ; similar considerations apply in the first case, only 

 as the lines are exceedingly narrow images of the slit, their 

 brightness, being independent of their distance from each other, 

 is inversely proportional to the length of the lines simply— ;>., 

 to the -uidlh of llie spectrum, having nothing to do with its 



Using the same notation as before, merely adding 



; = intensity of source of light. 



/ = length of the slit. 



a = linear aperture of the collimator object glass ; 

 and supposing the prisms and view telescope of a siz; to take in 

 the whole beam transmitted by the collimator, and that the 

 angular magnitude of the luminous object, as seen from the slit, 

 is sufficient to furnish a pencil large enough to fill the collimator 

 object glass, we shall then have for the quantity of light trans- 

 mitted to the prisms the expression 



i/u,'lL. 



This is afterwards diminished in passing through the prism 

 train and telescope. 



To estimate the precise amount of this loss is very difficult, 

 and the algebraic expression for it is of so complicated a 

 character that it would be of little use to attempt to introduce it 

 into our formula. Calling it .S', however (which of coarse is a 

 function of the number and refracting angle of the prisms, as well 

 as of the optical character of the glass), we may write for the 

 quantity of light effective in forming the spectrum, 



Q = i I -iv — ; - S. And this expression applies to both kinds 



of spectra — bright line and continuous. 



In the continuous spectrum this light is spread out over an 

 area whose length is the angular dispersion of the train * A , 

 multiplied by the magnifying power of the eye-piece and by the 

 focal length of the view telescope, and whose breadth is the 

 width of the spectrum. Putting A for this area, we have 



f 

 And for the intensity of light in the continuous spectrum, 

 which equals Q -r A, we get finally 



L ^ ilwa°--P?, 

 Ini' n Ayi -/• 



(4) 



I( we neglect the loss of light in transmission, nd takey'= y'', 

 the formula simplifies itself to 



L'= -^^^-> (5) 



m - n ti J 



Kither of these formu'.se shows how rapidly the light i-; cut 

 do.vn by any increase of the dispersive powtr, whether by adding 

 ti the prism train or by enlargement of the linear dimensions of 

 til*:; apparatus. 



Our only resource in dealing; with spectra of this kind, when 

 the limit of visibility on account of faintness is nearly attained, 

 seems to be either to increase ; or a. If the luminous object be 

 a point (like a star) we can do the former by concentrating its 

 light on the slit with a lens ; if it be diffuse, like the light of the 

 sky, I know no means for producing the desired concentration, 

 and we can only gain our end by increasing the angular aperture 

 of the collimator. 



For the discontinuous bright-line spectrum, the case is quite 

 different. Q, ;>. the quantity of light which goes to form the 

 spectrum, remains unchanged, but instead of A the whole area 

 covered by the spectrum we have only to consider its width, i.e. 

 the length of the lines, t 



• A = « (sin - • (mh X Sin i r) - Sin - > i>i Sin i r) ) where ^^ and 

 M , are respectively the indices of refraction for tfie lines A and H ; the prisms 

 being supposed to be so mounted as to maintain the position of minimum 

 deviation. 



t So long as the opening of the slit is small enough to secure accurate de- 

 finition of the lines, it is not necessary to take into account either this or the 

 magnifying power as diminishing the brightness of the lines by increasing 

 then- breadth, since irradiation alone gives them a sensible width sufficient 

 to render the effect of other causes comparatively unimportant. 



