VOL. XLVIII.] PHILOSOPHICAL TRANSACTIONS. 403 



m is the same power of m, as m is of* n; and therefore puts n = m', and n — m'. 

 Whereas, by all the experiments that have hitherto been made, the proportion 

 will come out thus, m — \ : n — l::m — m:« — n. 



The letters fixed on by Mr. Euler, to represent the radii of the 4 refracting 

 surfaces of his compound object-glass, are_/, g, h, k, and the distance of the 

 object he expresses by a ; then will the focal distance be = 



— y — iYT — 7x — xTI — x\ — i — TTT Now, says he, it is evident, that the different 



refrangibility of the rays would make no alteration, either in the place of the 

 image, or in its magnitude, if it were possible to determine the radii of the four 

 surfaces, so as to have n (^ — i) + jw ( j- — ;^ -j- f — f ) = n (f — i) -|- m ( j- — 

 ^ -}- X — i)- And this I shall readily grant. But when the surfaces are thus 

 proportioned, the sum of the refractions will be = O; that is to say, the emer- 

 gent rays will be parallel to the incident. For, if w (-^ — \) -\- m {f — 7 + i — i) 

 = N (t - i) + M (^ - t + i - i), then ,2-N(t-i)+m-M(t-t + 

 ■J- — i) = O. Also if ?i — N : m — M :: n — 1 : m—1, then n — 1 {f — i") + w— 1 

 (^ - t + i - i) = O; or otherwise „{f - i) + m {} - ^ + i - i) - ^ + 

 ■J- = O; which reduces the denominator of the fraction expressing the focal dis- 

 tance to X. Hence the focal distance will be = a ; or, in other words, the image 

 will be the object itself. And as, in this case, there will be no refraction, it will 

 be easy to conceive how there should be no aberration. 



And now Sir I think I have demonstrated, that Mr. Euler's theorem is entirely 

 founded on a new law of refraction of his own ; but that, according to the laws 

 discovered by experiment, the aberration arising from the different refrangibility 

 of light at the object-glass, cannot be corrected by any number of refractions 

 whatever. 



Lefter III.* From Mr. Euler to Mr. James Short, F.R.S. Dated Berlin, 



June 19, 1752. p. 292. 

 MoNS. 



Vous m'avez fait un tres sensible plaisir, en ayant dispose M. Dollond de re- 

 mettre la proposition de ses objections contre mes verres objectifs, jusqu' a ceque 

 j'y aurois repondu, et je vous en suis infiniment oblige. Je prends done la li- 

 berte de vous addresser ma reponse a lui, en vous priant, apr^s I'avoir dalgnee de 

 votre examen, de la vouloir bien lui remettre : et en cas que vous jugiez cette 

 matiere digne de I'attention de la Societe Royale, je vous prierois de lui commu- 

 niquer les preuves detaillees de ma theorie, que j'ai exposee dans cette lettre. 

 Cependant j'espere, que M. Dollond en sera satisfait, puisque je tombe d'accord 



* As these letiersr are on nice controveri>ial matters, it is considered safer and more satisfactory to 

 gire them in their original language. 



3 F 2 



