342 On the A.vial Dioptric System. 



In the particular case where ^^0 = it reduces to a pair of 

 coincident lines through C, whose intersection with AB is 

 the vertex of the line l^^^+m^r) = i). 



We note that if the coordinates of a point on AB are x^^yi, 0, 

 those of their vertex ^i, rj^, 0, and those of their base-point 

 Xi, Yi, 0, then 



and 



X,:Y. = ^,C'/-/'):yi(/-'')- • ■ • (23) 



From the symmetry of the transformation with reference 

 to the three double-points, A, B, C-, it is clear that in our 

 fundamental constructions we could interchange say B and C. 



Thus for every object-point there is a B-vertex and a B- 

 base-point lying in AC. 



From the purely mathematical point of view, the simplest 

 way to study analytically the transformation of Section III. 

 would be to begin with the equations (19) as defining the 

 transformation, and afterwards identify it with that defined 

 by the vertex-property. But from our present point of view 

 the course we have adopted seems better, especially as a pair 

 of double-points may be imaginary. 



Concluding Sote. 



In the foregoing study of the Axial Dioptric System and 

 its Cardinal Points, the problem of formulizing the calcula- 

 tions required for determining the positions of the Cardinal 

 Points, when the refracting surfaces and the media are given, 

 has been avoided. Euler, Lagrange, Gauss, and others have 

 applied continued fractions to such calculations. For practical 

 purposes possibly the direct graphic construction (on a 

 sufficiently large scale) of the images of two arbitrary extra- 

 axial object-points, treating the refractions successively, and 

 thereafter of the cardinal points (by the aid of the construc- 

 tions of Section I.) might be the most satisfactory method, 

 seeing that the data in practical cases are only approximately 

 exact. 



But however that may be, it seems to have been felt by 

 many writers on the subject that the avoidance of the some- 

 what complicated treatment by continued fractions would be 

 desirable in establishing the theory of these points. Clerk 

 Maxwell in his paper in the Quarterly Journal of Mathe- 

 matics, vol. ii., does this by assuming the existence of a 

 '' perfect optical instrument" fulfilling certain conditions, 

 and a considerable resemblance will be found between some 

 parts of the present paper and of that of Clerk Maxwell. 



Mobius, again, who wrote on the subject both before and 



