486 Prof. J. Loudon on Geometrical Methods in the Theory 



trical method for determining the point conjugate to any 

 given one. 



The points R, W from which distances are measured, it is 

 to be observed, are any two conjugate points, such, for ex- 

 ample, as the principal points, or nodal points ; and they may 

 in particular cases coincide when they are self-conjugate. 



It is proposed in the following paper to employ the method 

 indicated chiefly in discussing certain propositions in the 

 theory of thick lenses. 



I. 



2. In the case of refraction at a single spherical surface, 



where/,/ 7 are the distances of the primary and secondary 

 principal foci F, W , 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 (Plate X.), 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 



ff = dd f , 

 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 7 are conjugate points, as also Q, Q', then drawing 

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



dd'=(d+-?q){d f -v'q f ), 



which reduces at once to 



pq + fq' 



This is of the form 



j) + jy =1 ; ...... (2) 



where the distances d, D are measured from P, and d' } D' 





