1880.] of a single lens for parallel rays. 375 



If ;a = 2, (8) gives 



-hf • ^ = --7- 



^J • f 16' 



not yV. a.s stated by Coddington. The aberration tends indeed to 

 become less as /x increases, but it remains considerable for all 

 substances known in nature. 



It seems to have been thought evident that great advantage 

 would result from higher refracting power on accouut of its allow- 

 ing the use of more moderate curvatures. It appears however 

 from (5) and (6) that as jju increases, r and s do not tend to become 

 infinite for the form of minimum aberration, but approach the 

 finite value /. 



(2) A. Freeman, M.A., Note on the value of the least root of an 

 equation allied to J^ {z) = 0. 



The equation in question, viz. 



-| **-* tAj ^A/ On /I \ 



-'- T2 ~r -| 2 92 -1 2 92 02 "T" -12 92 02 A2 '^^^ ^ l-*-/ 



is identical with J^ (z) = if we write ^ for w. 



(See Lommel's BesseVschen Functionen, page 26.) 



In a memoir on the Calculus of Variations printed in the 

 twelfth volume of Memoires de V Academie des Sciences, Paris, 

 1833, Poisson had occasion for the least root of the equation 

 above given. He seems to have applied to M. Largeteau of the 

 Bureau des Longitudes, by whom the calculation was made, and 

 Poisson quotes the number thus : 1"46796491. 



The equation occurs in the problem of the flow of heat in an 

 infinite cylinder, and in my edition of Fourier's Theorie Analytique 

 de la Ghaleur I gave, in a note, page 3i0, the result of an approxi- 

 mation, including terms of the series as far as x^ ; my result was 

 1-4467. 



Suspecting some error in the Poisson-Largeteau number, I first 

 supposed it might need only the insertion of an additional figure 

 4 after the decimal point, taking it to have really been 1'446796491. 



On substituting this value of x in the first five terms of the 

 series, the algebraic sum became — 0*000009, but the value of the 

 fifth term was as large as '007607. 



