470 Lord Rayleigh's Notes on some 



The first advance upon the position attained by Smith is to 

 be found in Kirchhoff's celebrated (but perhaps little read) 

 memoir, " Ueber das Verhaltniss zwischen dem Emissions- 

 vermogen und dem Absorptionsvermogen der Korper fur 

 Warme und Licht."^" The Theory of Exchanges renders it 

 evident that the law of apparent brightness must have an even 

 higher generality than Smith had claimed for it. No limita- 

 tion can be admitted to systems of optical surfaces centred 

 upon an axis, to which the rays are supposed to be but slightly 

 inclined. Kirchhoff's investigation is founded upon Hamilton's 

 characteristic function T, which we may here take to represent 

 the reduced optical distance (\fjuds) along a ray between any 

 two points. At the extremities 0, 0', of the central ray of a 

 pencil undergoing any number of reflections and refractions, 

 planes are drawn perpendicular to the final directions of the 

 ray, and in these planes rectangular coordinate axes a? 1? y 1} 

 x 2 , 2/2 are taken. T expresses the reduced distance between 

 a point # 1; yi, in the first plane and a point x 2 , y% in the 

 second plane, as a function of these four variables. In Smith's 

 terminology Kirchhoff's result may be thus stated : — The in- 

 verse square of the apparent distance between and 0' is 



cPT d 2 T d 2 T d 2 T 



dx x dx 2 dyi dy 2 dx\ dy 2 dx 2 dy^ 



the apparent distance being the same whether 0' be seen 

 from 0, or be seen from 0'. 



the foundation of the method long since used by English opticians 

 for determining the magnifying powers of telescopes of all kinds, which 

 form an image of the object-glass beyond the eye-glass, by measuring 

 the diameter of that image. The author hazards, in this paper, the 

 very singular assertion, that the illumination of the object must 

 be the same in all telescopes whatever, notwithstanding the common 

 opinion that it depends upon the magnitude of the object-glass ; and his 

 reasoning would be correct, if the pupil of the eye were always less than 

 the image of the object-glass in question; since, as he observes, the 

 density of the light in this image is always inversely as the magnifying 

 power ; but he forgets to consider that the illumination on the retina, 

 when the whole pencil is taken in, is in the j oint ratio of the density and 

 the extent ; a consideration which j ustifi.es the common opinion on this 

 subject, and shows that a most profound mathematician may be egre- 

 giously mistaken in his conclusions, if he proceeds to calculate upon 

 erroneous grounds. It deserves, however, to be remembered that the 

 brightness of any given angular portion of a magnified image must 

 always be somewhat less than that of an equal portion of the object seen 

 by the naked eye, because it can be no greater if the pencil fills the pupil, 

 and will be less in proportion as the pencil is smaller than the pupil, 

 besides the unavoidable loss of light at the refracting surfaces." 



In his Theorie des Lunettes, Berlin, 1778, Lagrange himself refers to 

 Smith's Optics, Oh. v. Book ii. 



* Pogg. Ann. t. cix. p. 275 (1860). 



