8 PROCEEDINGS OF THE CANADIAN INSTITUTE. 
where f, f’ are the distances of the primary and secondary principal 
toci 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 (f, /’), 
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 
O, FO being equal to FX. 
3. From similar triangles PFX, XF’P’, it is immediately seen 
that 
If' = dd, 
where PE: — da, 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. |. 
4. If P, P’ are conjugate points, as also Q, Q’, then drawing 
PXP’, QXQ’, as in Fig. 1, we have 
dd’ = (d + PQ) (d’ — P’'Q’) 
which reduces at once to 
d d 
— 9 + Pg = 
This is of the form 
d @ — 1 ef (2) 
DTD Lie ol ue 2 
where the distances d, D are measured from P, and d’, D’ from its 
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 « 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 O, the centre of the 
sphere, the relation (2) becomes 
Lexie 
pe ae 
where (Fig. 3) OF — /’, OP = p, ke. 
As in § 3 we have dd = f'f. 
=" 
