184 CALIFORNIA ACADEMY OF SCIENCES [Proc. 4th Ser. 



This is the final shape of the spreading pencil according to 

 Sturm's theory. 



The actual focal transformations are much more compli- 

 cated as will be comprehensible from a study of the second 

 set of diagrams. 



(a) The initial phase precisely like that used in the pre- 

 ceding series of diagrams. 



(b) The first conjunction, the plane where points 1 and 2 

 are nearest. There is a point in line between 2 and 5 which 

 does coincide with 1 on a plane not far above this. 



(c) The first crossing, the plane where 4 coincides with 5. 

 The point 4 has been travelling along the line 4 and 5 and 

 when it reaches that point the remainder of the line stretches 

 out a short distance behind. 



(d) The second conjunction being the plane showing the 

 nearest approach of 1 and 3. This phase is probably in general 

 shape mose nearly comparable with phase II, according to 

 the Sturm theory. 



(e) The second crossing, in which plane points 4 and 6 

 coincide. This phase probably could be chosen as the nearest 

 approximation to the third phase of the old theory. 



(f) The first marginal focus at which the points 1 and 7 

 coincide. It is only the beginning of the coincidence of the 

 line 1, 4, 7. 



(g) The third conjunction at which points 2 and 3 are 

 nearest. This diagram also represents the completion of the 

 reversal of line 1, 4, 7. 



In this and the following diagram the constriction between 

 the erect and inverted portion of the image is by no means as 

 sharp as here shown where the lines indicate the edges only 

 of the infolded image, and the whole image at the point of 

 crossing has a very appreciable width, still the general shape 

 of the pencil section conforms very closely with these diagrams. 



(h) The third crossing where points 5 and 6 coincide but 

 it is not the conclusion of the reversal of the line, because the 

 points on the line 5, 6 are spread out behind this point. This 

 and the preceding five phases include the region in which the 

 reversal of the rays is brought about as is supposed to occur 

 in the meridianal locus. 



