22 Professor Airy on the Spherical Aberration 



will be seen that all the rays of the pencil after refraction pass 

 through different parts of the line Ee. Now, let Jf be another ray 

 near AF, and also in the plane of the paper (which we shall call 

 the paper plane), and let it be refracted in the direction ge. It is 

 evident that GE and ge converge to the point M: and it will easily 

 be seen that all rays in the paper plane, or in a plane parallel to 

 the paper plane (the breadth of the pencil being small) converge 

 io a point in a line drawn through M perpendicular to the plane 

 of the paper. That is, all the rays of the pencil pass through a line 

 perpendicular to the paper at the point M, and all pass through the 

 line Ee in the plane of the paper. And it is plain that all the rays 

 of the pencil do not converge to any one point whatever. 



In the same manner it may be understood, that when a pencil 

 of rays diverges from a point L, Fig. 4, which is not in the axis 

 of a lens, all the rays after refraction may pass through one straight 

 line perpendicular to the paper at some point S, and through 

 another straight line in the paper plane, as Mm. That this is 

 the case, our investigation will shew. The mode, therefore, of 

 discovering the course of a pencil of rays after refraction is to 

 examine the position of the two lines through which, after re- 

 fraction, all the rays of the pencil must pass. 



I shall first remark, that in Fig. 3, we might, without sensible 

 error, have asserted that all the rays of the pencil pass thi'ougli 

 the line Eh- (which is perpendicular to the axis of the lens) instead 

 of the line Ee. For the only error is this: that we assume the 

 convergence of the rays in the perjjendicular plane passing through 

 4re to take place at k instead of at e. But the interval eA: is pro- 

 l>ortional to the breadth of the pencil, and we shall always neglect 

 quantities of this order in comparison with quantities, as EM, 

 independent of that breadth. The same reasoning will apply to 

 other cases: and we shall, therefore, without further explanation, 

 assume that a pencil always converges to two lines, one perpen- 



