66 Astronomical Society. 



A paper was read " On double object-glasses ; by M. Littrow." 



The first part of this communication is devoted to the derivation 

 of the equations expressing the conditions proposed by Mr. Her- 

 schel for the destruction of the aberration of sphericity in a thin 

 double object-glass, by the ordinary processes in use among the 

 German geometricians for such investigations. The author states 

 himself to have entered on this investigation with a view to disse- 

 minate a knowledge of the theory alluded to in his own country ; 

 but being induced thereby to resume some former investigations of 

 his own, he takes the opportunity to communicate to the Society 

 his own principal results. 



Taking for granted the well-known expression of Euler for the 

 aberration of a lens of two surfaces, and developing it in descending 

 powers of the distance of the radiant point from the lens, he obtains 

 expressions, from which, by proper management, and substitution 

 of similar quantities for a second lens, he derives the two final equa- 

 tions (A) and (z) demonstrated by Mr. Herschel. He then pro- 

 ceeds to the main object of his paper. This may be briefly stated 

 to be the embodying of the relations expressive of the refraction 

 of a ray through any four spherical surfaces, however situated, 

 (provided they have a common axis), in trigonometrical equations, 

 in which no quantity is regarded as small, and of course nothing 

 neglected. These equations are in themselves sufficiently simple 

 when undeveloped, and in that state may very readily be applied 

 to determine whether any proposed construction of an object-glass 

 really does satisfy the essential geometrical conditions of a perfect 

 telescope, by producing a rigorous union of different coloured rays, 

 and rays incident on different parts of the object-glass. Accordingly 

 the author instances their application to a construction recently 

 proposed by a German optician, as of peculiar excellence. In this 

 construction the indices of refraction of the crown and flint lenses 

 being respectively 1*53 and 1*60, and their dispersive ratio 0*25, the 

 thickness of the crown lens 0-01 , that of the flint 0, and the lenses 

 being supposed in contact, the radii of the surfaces are 



For the crown 1st surface (convex) 0*69281 



2nd surface (convex) 2-255319 



For the flint 1st surface (concave) 1-543030 



2nd surface (concave) 5*768005 

 Substituting these data in his equations he finds that they satisfy 

 sufficiently well the conditions of achromaticity, but that they are 

 very far from being entitled to the same encomium when the sphe- 

 rical aberration is considered. 



But when the question is inverted, and the problem is, not to try 

 whether a proposed construction be good or not, but, a priori, to 

 determine what is best, the equations in question, though simple 

 enough in their trigonometrical form, become complicated by the 

 algebraic developments their direct analytical resolution necessi- 

 tates. Now the essence of M. Littrow's proposed method is to do 

 away with all this development, so far as it tends to produce com- 

 plication ; and after preparing the equations in the most convenient 

 manner the case will allow, to substitute for their direct algebraical, 



