316 Dr. S. P. Thompson on 



menon, together with the curves for the E.M.F. of various 

 couples during heating and cooling- — thermo-electric hysteresis 

 curves as they may be called*. It is very probable that the 

 peculiar thermo-electric deportment of iron, and some of the 

 alloys of iron described in this paper, is intimately associated 

 with the phenomenon of recalescence, or rather of the series 

 of recalescent points which exist in iron and steel. 



XXVII. On Obliquely -crossed Cylindrical Lenses. 

 By Silvanus P. Thompson, f>.Sc., F.R.S.\ 



C CYLINDRICAL lenses have not claimed much attention 

 J from writers on geometrical optics. Certain of their 

 properties, which make them invaluable to the ophthalmic 

 surgeon for the correction of astigmatism, are, however, 

 considered in all modern treatises on ophthalmics. Airy 

 applied them for this purpose, but Donders first treated 

 them systematically, in 1862, in his w T ork Astigmatismus 

 und cylindrische Gltiser. Reusch in 1868 published his 

 Theorie der Cylinderlinsen. Javal has written much on 

 the subject from the ophthalmological point of view, as has 

 Mr. S. M. Burnett. Mr. C. F. Prentice has also written of 

 them in two works, a ' Treatise on Ophthalmic Lenses ' and 

 ' Dioptric Formulas for combined Cylindrical Lenses.'' 

 Stokes + has proposed a cylindrical lens of variable power by 

 combining at a variable angle two equal cylindrical lenses of 

 opposite sign. The problem of the optical properties of 

 crossed cylindrical lenses was touched upon by Reusch and 

 by Donders. Kriiss has written upon the aberration of 

 " bicylindrical " lenses, meaning by that term such lenses as 

 have both surfaces cylindrical and of equal curvature, but 

 with their axes mutually at right-angles. Lastly, Prentice 

 has given very elaborate rules for calculating the equivalent 

 sphero-cylindrical lens for any combination of two lenses 

 crossed at any angle. The importance of the solution of this 

 problem arises from a point in modern ophthalmic practice: — 

 In cases of ordinary astigmatism in which the refractive 



* As was pointed out by Professor G. F. FitzGerald, F.T.C D., F.R.S., 



at the meeting of the Society when this paper was read, the thermo- 

 electric hysteresis here referred to is, no doubt, the cause of the thermo- 

 current which is produced in an iron wire steadily moved through 

 a flame, a phenomenon first noticed and investigated "by Dr. F. T. 

 Trouton, F.P.S. See Proc. Roy. Dublin Society, March 1886. I am also 

 greatly indebted to Professor FitzGerald for other suggestions he has 

 made in reading the proof of this paper. 



t Communicated by the Physical Society : read Dec. 8, 1899. 



X Mathematical and Physical Papers, vol. ii. p. 172. 



