b PROCEEDINGS OF THE CANADIAN INSTITUTE. 



■wherej^,/' are the distances of the primary and secondary principal 

 foci F, F', and p, p the distances of the object and image P, P', from 

 A, the point where the principal axis meets the sphere. 



Let the standard case be that of refraction into a denser medium ^ 

 whose surface is convex, the direction of the light being from left to 

 right. Then drawing axes AF, AF', and taking the point X {/,/'), 

 as in Fig. 1, we see that the point conjugate to P on one axis is- 

 the intersection of PX with the other. 



It appears from the figure that A is a self-conjugate point, as also- 

 0, FO being equal to FX. 



3. From similar triangles PFX, XF'P', it is immediately seen 

 that 



//' = dd', 

 where PF = d, P'F' = d'. 



If the rule of signs (§1) be applied to the measurement of d, d'' 

 on the two axes,^it is to be observed that they are of the same sign, 

 both being negative, for example, in Fig. 1 . 



4. If P, P' are conjugate points, as also Q, Q', then drawing 

 PXF, QXQ', as in Fig. 1, we have 



dd' = {d + Vq) {d' — P'Q') 



whicli reduces at once to 



d d' 



~ "PQ "^ FQ ' ^ ■ 



This is of the form 



d d' , 



^+^,= 1 ....... (2) 



where the distances d, D are measui'ed from P, and d', D' from it& 

 conjugate P', the rule of signs being that already referred to in § 1 . 



5. Fig. 2 exhibits the construction adapted to formula (2). P in 

 the X axis coincides with its conjugate P' in the y axis, and the line 

 joining any other two conjugate points (Q, Q') on the two axes 

 passes through the point {d, d'). 



If the origin be the self-conjugate point 0, the centre of the- 

 sphere, the relation (2) becomes 



^- + 4 = 1, 



P P 

 where (Fig. 3) OF =/', OP =^;, &c.. 

 As in § 3 we have dd' ^f'f. 



