of Refraction at one or more Spherical Surfaces, 493 

 cident,/ is negative,/' positive; and formula (1) becomes 



P P 

 Hence the line joining conjugate points on the two axes 

 passes through X(— /,/), as in fig. 11. 

 For a concave mirror the formula is 



P P' ' 



and X is (/, — /), as in fig. 12. 



25. In either case we have, from the similar triangles 

 PFX, XFP' (fig. 11 or 12), 



PF_FX 

 FX ~ FF ; 

 that is 



which is Newton's formula. 



If d and d! be measured from P and P' in accordance 

 with the rule of signs § 1, this formula should be written 



dd'=-r, 



as also appears by deducing it from the relation dd'—ff 

 of § 3. 



26. The relation between the lengths of the object and 

 image is most readily obtained by making the axes cross at 0, 

 the centre of the mirror. 



Thus, for a convex mirror we have (fig. 13) 



a/ _ OF FX _ / 

 co ~ OP ~~ PF ~ d" 



In the case either of a convex or a concave mirror it may 

 be remarked that, if account be taken of the signs of/,/ r , 

 d. d\ the relation 



co'd-f 



determines whether the image is erect or inverted, the sign of 



r 

 — being positive in the former case, and negative in the latter. 



III. 



27. Since writing the above, it has occurred to me that the 

 relation dd! =ff leads to two other simple geometrical methods 

 for exhibiting the relations between the conjugate points. 



Thus, if we separate the two axes FF 7 , FF so that F in the 



