Geometrical Methods in the Theory of Refraction. 485 



hammering. The reason of the combination of the carbon 

 would, however, be the same in both cases — namely, the closer 

 approximation of the molecules produced by pressure ; and the 

 smaller effect produced by hammering is explained by the 

 fact that, in the first place, upon hammering the approxima- 

 tion takes place in one direction only, but upon cooling in 

 all directions at once, and that, secondly, the pressures exerted 

 are less intense than those produced by cooling. 



If this view of the cause of the hardening of steel produced 

 by sudden cooling be correct, it must also be possible to 

 harden steel by allowing it to cool slowly from the red-hot 

 condition, but so that during the cooling it is exposed to high 

 pressure. Steel thus treated ought, then, like tempered steel, 

 to contain more combined carbon, and to be hard. In fact, 

 according to Clemandot * and Lan | ; both of these conclusions 

 are verified by experiment. 



LV. Geometrical Methods in the Theory of Refraction at one 

 or more Spherical Surfaces. By James Loudon, University 

 College, Toronto^. 



[Plate X.] 



1. TN cases of reflection or refraction at a spherical surface, 

 -«- or a combination of spherical surfaces, or lenses, if 

 F, F' be the primary and secondary principal foci of the 

 surface, lens, or combination, and (P, P'), (R, R') pairs of 

 conjugate points, it is known (§ 6) that 



^ + ^ = 1, (1) 



p p v ' 



where /=RF,p = RP, /' = R'F', / = R'P'; and where the 

 positive direction from R for / and p is opposite to, whilst 

 that from R' for/' and p f is the same as, the direction of the 

 incident pencil. 



Now since the relation (1) expresses the condition that the 



OS II 



line - + —, = 1 passes through the point (/,/'), it follows that 



if the coincident lines FRR'F', FR'RF' be separated so that 

 R on the first axis coincides with R' on the second, the line 

 joining P on the former to P' on the latter will always pass 

 through the fixed point (/,/'). Hence we derive a geome- 



* Comptes Eendus, xciv. p. 703 (1882) ; xcv. p. 587 (1882). 



t Ibid. xciv. p. 952 (1882). 



X Communicated by the Author. 



