Lord Rayleigh's Investigations in Optics. 179 



and 



•/•=**•, (3) 



the usual formula. 



When the incidence is oblique, there are different foci in the 

 primary and the secondary planes. Considering first the case 

 of the primary plane, let A C B Fig. 9. 



(fig. 9) be the mirror, and F the 

 focus. AD = y, CD = t, HCD = <£. 

 A L is the course which the ray 

 GrA would take if reflected by 

 the plane surface A D B. C is 

 that point of the mirror at which 

 the tangent is parallel to AB. 

 The retardation at F of the ray 

 G A F is A F— A L ; and the re- 

 tardation of the ray H C F due to the curvature of the mirror 

 is 2t cos <f>. These retardations must be equal ; and thus 



AF-AL= v/(AL 2 + FL 2 )-AL = 2*cos<£. 



When the aperture is small, V(AL 2 + FL 2 )— AL is approxi- 



FL 2 

 mately J tt~' ultimately AL may be identified with f u the 



focal length in the primary plane, and FL may be identified 

 with y cos <£, so that 



t/ 2 cos 2 <f> ft . . /iN 



*-yr* = 2tca8<l>, (4) 



or 



, _ y? cos eft ,. 



fl -~4FT {) 



Thus it appears that, so far as the primary focal length is 

 concerned, the diminished retardation of the central ray due 

 to obliquity is outweighed by the corresponding diminution of 

 effective aperture (FL) ; but although in consequence of ob- 

 liquity a greater change of focus (estimated from that required 

 for the plane surface) is necessary in order to get the best 

 result, still, if no change of focus be admitted, the error due to 

 curvature is less sensible in oblique than in direct reflection. 



The preceding discussion assumes that the same extent of 

 surface is used in all cases. In testing planes by reflection it 

 often happens that a greater extent of surface is used in the 

 case of obliquity, the field being limited by the object-glass 

 of the telescope rather than by the reflecting plane. Under 

 such circumstances the loss of definition (with focus unaltered) 

 due to curvature is aggravated by obliquity. 



