[From The Quarterly Journal of Pure and Applied Mathematics, Vol. II.] 



XVII. On the General Laws of Optical Instruments. 



THE optical effects of compound instruments have been generally deduced 

 from those of the elementary parts of which they are composed. The formulae 

 given in most works on Optics for calculating the effect of each spherical sur- 

 face are simple enough, but, when we attempt to carry on our calculations from 

 one of these surfaces to the next, we arrive at fractional expressions so com- 

 plicated as to make the subsequent steps very troublesome. 



Euler (Acad. R. de Berlin, 1757, 1761. Acad. K. de Paris, 1765) has attacked 

 these expressions, but his investigations are not easy reading. Lagrange (Acad. 

 Berlin, 1778, 1803) has reduced the case to the theory of continued fractions 

 and so obtained general laws. 



Gauss (Dioptrische Untersuchungen, Gottingen, 1841) has treated the subject 

 with that combination of analytical skill with practical ability which he displays 

 elsewhere, and has made use of the properties of principal foci and principal 

 planes. An account of these researches is given by Prof. Miller in the third 

 volume of Taylor's Scientific Memoirs. It is also given entire in French by 

 M. Bravais in Liouville's Journal for 1856, with additions by the translator. 



The method of Gauss has been followed by Prof. Listing in his Treatise 

 on the Dioptrics of the Eye (in Wagner's Handworterbuch der Physiologic) from 

 whom I copy these references, and by Prof. Helmholtz in his Treatise on 

 Physiological Optics (in Karsten's Cyclopadie). 



The earliest general investigations are those of Cotes, given in Smith's 

 Optics, n. 76 (1738). The method there is geometrical, and perfectly general, 

 but proceeding from the elementary cases to the more complex by the method 

 of mathematical induction. Some of his modes of expression, as for instance his 

 measure of "apparent distance," have never come into use, although his results 

 may easily be expressed more intelligibly ; and indeed the whole fabric of 



