of Lippich's in Geometrical Optics. 119 



which agree with the expression found for the projection of! 

 the normal at P. Hence the normal at P projects into the 

 normal at the projection of P. 



The condition that the projections (3) should be conjugate, 

 or form part of the same ray of light, is 



//L'-yaL^'M'-^M, 

 ~1 by' 



which is true by (1). Thus the theorem is proved for the 

 tangential section, whatever the values of L and M may be. 

 If the projection had been upon the plane j/ = 0, the 

 condition obtained for conjugacy of the projected paths 

 would have been 



p 



fiL \/L' 2 + W 2 VI^ + N 2 



or neglecting N 2 and N' 2 , 



+ / '4- >> " L ' ± L 

 ±{ji ±fl) = —, , 



which does not agree with (1) generally. The expressions 

 agree if L' = L=+1, i.e. if the ray is parallel to the 

 normal at 0. 



If the chief ray is normal, then evidently it is immaterial 

 which of the two principal sections is regarded as the tan- 

 gential section. The ray lies in both. 



II. Projection on the Sagittal Sections. 



The general equations (see Note at end of discussion) of 

 the projection of a straight line 



x — x'_ y—y' _z — ~ 



on a plane Ix + my + nz—p (p being the perpendicular on the 

 plane from the origin) are 



x — x' + 1(1% + my' + nz' — p) _y —y' + m(lx' -f my' + nz —p) 

 L — £cos# M — mcosO 



_ z — z'-\rn(lx' + my' + nz — p) 

 N— 72COS 6 



6 being the angle between the straight line and the normal 



