104 Dr Searle, Experiments with a plane diffraction grating 



radius. If Q^Ri, drawn perpendicular to Q-fi in Fig. 3, cuts OC^ 

 in i?i, the radius of curvature of the caustic at Q^ is ^QJH^. The 

 length of the caustic between the points of contact of the tangents 

 from Ml, iVi, where M^N^ is the width of grating actually used, is 

 3 (^iMi ~ ^1^1 ) approximately. 



§ 7. Adjustment of the lens. The lens, a converging system, is 

 adjusted optically. Let its focal length be / and the distance 

 between its nodal points be t, where t is positive when the distance 

 between the principal foci exceeds 2/; for a projection lens as shown 

 in Fig. 3, t will be negative. When the distance of the luminous 

 point E from the scale ACB exceeds 4/+ t, there are two positions 

 of the lens for which an image of E is formed on the scale. Let 

 M, N (Fig. 4) be the nodal points corresponding to the principal 



Fig. 4. 



foci to the right and left of L. In Fig. 4 let EC be the horizontal 

 line through E parallel to the bench and let the other Unes be pro- 

 jections upon the horizontal plane through EC. Let MR, NS be 

 the perpendiculars from M, N on EC. Let RE = p, SO = q. Then 

 in the second position of the lens, R'E = q, S'C = p. When the 

 angles are small, RS = R'S' = t. Let I, I' be the images of E in 

 the two cases. 



Take CE as axis of a; and horizontal and vertical lines through C 

 as axes of y and z. Let the second and third coordinates of M, N, 

 I, r be y, z, 7), I, r, Z, Y' , Z' . Since /iV is parallel to ME, and 

 I'W to M'E, 



7 = 7y + yq\p, Z = ^ + zqjp, T' = t^ + yp\q, Z' = ^ + zp\q. 



Since {p + q)\pq = II f, 



Y-Y' = y{q-p)lf, Z-Z' = z{q-p)lf. 



Hence, if the image has the same position for both cases, then 

 y = 0, and 2 = 0, and thus M lies on EC. The emergent ray 

 through N will then be parallel to EC for all positions of the lens 

 carriage. 



The lens is best mounted so that it can turn about a vertical 

 pivot whose axis passes through M. If the support to which the 

 pivot is fixed is moved through the distance MR at right angles 

 to the bench, M will be brought on to EC. Then by turning the 

 lens about M, N can be brought on to EC. 



