M. Biot on certain Points of Mathematical Optics. 481 



After having developed this important application, I re- 

 sume the general formulae peculiar to dioptric systems, and I 

 employ them to establish the theory of eye-pieces applied to 

 achromatic object-glasses. I limit them to the usual case, in 

 which the component lenses of these eye-pieces are made of 

 glasses of the same kind, and I deduce from thence the 

 exact rules for their construction, as well as all the peculiari- 

 ties of their effects. I then apply the same formulae to the 

 analysis of the night glasses employed as finders, to that of the 

 day or long-sight glasses, in which the objects are seen erect, 

 and I deduce the best conditions for their construction. I 

 finally employ them in the discussion of the heliometer, which 

 has latterly acquired celebrity from the use which M. Bessel 

 has made of the one constructed by Fraunhofer for the obser- 

 vatory at Kcenigsberg. I deduce from them the rigorous ex- 

 pression of its effects, as well as the modifications which they 

 necessarily undergo under the influence of different tempera- 

 tures, — a consideration essential for the appreciation of the 

 degree of accuracy of the extremely delicate measures which it 

 is intended to furnish. I then concurrently expound the pro- 

 cess of duplication which M. Arago has devised for measuring 

 small visual angles between celestial objects, and I give it with 

 the latest improvements which he has recently introduced. 



I am aware that so extended an exposition of the theory of 

 optical instruments may appear too irrelevant to a special 

 treatise on astronomy to be included in it; but I have been 

 constrained to this necessity, when, desiring to present a suc- 

 cinct but exact analysis of the effects of these instruments, and 

 of the principles by which we may regulate and rectify them, 

 and appreciate their qualities or their defects, I have ob- 

 served that, after so many mathematical investigations upon 

 this subject undertaken by the most able geometricians, we 

 did not yet possess, even in the case of very small inflexions, 

 any analytical method which presented the definitive effects 

 of optical spherical systems, under a general and explicit form, 

 in which we had only to substitute numbers to appreciate their 

 results. So that we were obliged to establish for each instru- 

 ment a particular discussion, founded upon special simplifica- 

 tions, of which, for the m'»st part, we could not appreciate the 

 degree of exactness, still less justify the necessity. The only 

 advances hitlierto made toward this object, and which, with- 

 out having completely attained it, offered at least an expecta- 

 tion of the possibility of arriving at it, were, I think, the beau- 

 tiful theorems of Cotes, some generalized inductions of Euler, 

 and, above all, the remarkable indications which Lagrange 

 had given of the employment of finite differences to express 



Phil, Mag. S. 3. Vol. 26. No. 175. June 184.5. 2 K 



