MR. DOLLOND ON A CONCAVE ACHROMATIC LENS. 203 



at pleasure, and compute its aberration for rays converging to the distance d, which 

 may be done by the following formulae, a being the index. 



"Find 



,, (a + 1) a d d! , , r 



"^ =^j::r-rdr, b = j^^, -jj = c, -p- = c', and^=y; 



then the aberration will be 



aberration - J -^^±^ v ^ + (^ + 2) g (^^ + 1)^ {c'+2-b)q -] a 

 abeiration - | («, _ ^y X ,^ac' + a+ i)^+ (T?TTr ^ ? J ^27 



" Let the quantity when found be called m, then for the flint lens proceed as below, 

 the radii being r ', r "', the latter towards the eye, and the index a. 

 " Find 



, __ {a! -{■ l)2dr"> _ a' Qd __ d' __ , j ^' _ 



^~ 2«'rf — r"' ^ " "" a' + P r"' ~ ^' ^w ~ ^ , and p^ — g'. 



" Then find r ", /' and §-, such, that 



SS'J±_9Y ^ c + {a + Q)q (d + If (^c' + 2-b)g -] ^ _^ __ 



I {a<c - qf ^ c{a'd + a' + if "i" (6c' + Jf ^ d J ^ 2r"' "" ^^ 



and the resulting curves will be those required. 



" To produce this latter equality is the only difficulty in the operation, and to treat 

 it as a common equation would lead to immense labour. I have therefore always 

 contented myself with pursuing the more simple method of trial and error, its facility 

 fully compensating, in my mind, for its want of scientific elegance. 



" It may be proper to observe, that I proposed the lens to double the magnifying 

 power, and the curves were computed accordingly, but the formulae will of course 

 apply to magnifying in any ratio. 



" I hope this explanation will be found intelligible, and I am pleased to find my 

 proposition has been found useful. 



" I remain, dear Sir, 



" Yours very truly, 



(Signed) " Peter Barlow." 



I have only to add to the foregoing relation of facts, that I do hope they will prove 

 satisfactory to those friends who have felt so much interested upon the subject as to 

 induce me to write this Paper, it not being my wish to take credit to myself for any- 

 thing like an invention, but merely for the application of the lens to the micrometer, 

 as I am fully convinced that a concave lens, either simple or achromatic, was never 

 so applied before. 



* See Philosophical Transactions, 1827, p. 244. 

 Fehriiary 17 th, 1834. 



2 D 2 



