476 Dr. C. V. Drysdale on the 



thickness at its centre being £, which is the sag corresponding 

 to the total curvature of the lens. A wave passing through 

 it will have its centre retarded by an amount s = (//,— l)t as 

 with a thin prism. I£ the wave is initially plane it receives 

 a curvature corresponding to the sag S, which we term the 

 convergence F of the lens, and we therefore have immediately 



F = (/Lt — 1)0 corresponding to S = (yu, — l)a for a thin prism. 



If the wave is initially curved or convergent by an amount 

 U, the retardation increases its convergence by the amount 



Y-U = ( A 6-1)C = F. 



The thickness t of the lens can be made up of the two sags 

 t and t 2 to the curvatures E and E 2 of the faces. 

 Hence y_ U = F = ^_i)c = (^_i)(R _R 2 ). 



By considering a pinhole stop in contact with the lens we 



v U 



immediately have m = - = ^ as with the mirror ; and the 



properties of lenses and formation of images follow 



immediately. 



Thin Cylindrical Lenses. — For a principal section of a 



cylindrical lens perpendicular to its axis, we have obviously 



the same relation as for a spherical lens. For an oblique 



section the retardation of the wave-front is obviously the same, 



but the breadth of the beam is increased (fig. 4) . From the 



2h 

 spherometer formula R = — g- we have 



R'=R/-,Y=Rsin 2 0, 



"-GT- 



and hence also F' = F sin 2 0. In a meridian perpendicular to- 

 the former we must similarly have F // = Fcos 2 ^, hence 

 F' + F // = F, or the sum of the convergences in two meridians 

 at right angles is constant and equal to the convergence of 

 the lens, analogous to a well-known theorem for curved 

 surfaces. 



Deviation and Decentring of Lenses. — By multiplying the 

 equation V — U = F by the height of intersection of the pencil 

 with the surface we have immediately 



tan cr 2 — tancr 1 = F/i or <t = ct2 — cr^F/i 



if the angles are small. This gives us the valuable result that 

 the deviation at any zone of a thin lens in prism-dioptres is 

 given by the product of the convergence of the lens in dioptres 



