Vol. IX] WOODWORTH— OPTICS OF THE MICROSCOPE 185 



(i) The median focus being the coincidence of points 2 

 and 8. This point is probably the nearest to the sagittal locus 

 of Sturm's system, and here also the shape of the pencil most 

 nearly suggests the fourth oblong phase. 



( j ) The optical focus where the reversal is complete except 

 for the last line 3, 6, 9. This is approximately the plane which 

 will be chosen by the eye as the focus. 



(k) The second marginal focus at which 3 and 9 coincide. 

 This and the preceding five phases intersect the locus obliquely. 



(1) The anal phase, showing the complete transformation 

 and approximate form of the pencil. 



A striking and significant difference between these two sets 

 of diagrams is that in the first rays 3 and 9 intersect nearest 

 the set of refracting points and in the other diagram this 

 intersection is furthest from that surface. The further fact 

 that the loci of which these intersections are one limit, are 

 nearly at right angles with one another emphasizes the irre- 

 concilable differences between these conceptions. 



The oblique locus in the case of a reading glass is not a 

 line but a very narrow linear figure, appreciably curved at the 

 ends, oblique relative to the optical axis and approximately 

 normal to an equivalent single lens surface. 



These diagrams exhibit only two of the five aberrations. 

 Distortion and curvature are concerned with differences be- 

 tween the foci of the different portions of the field. Axial 

 aberration is that which appears when the lens is normal to 

 the beam of light. The aberrations here exhibited are 

 astigmatism and coma. Considering point 5 the middle of 

 the light pencil then, the 1, 4, 7 region represents astigmatism 

 and the 3, 6, 9 region represents coma. The optical focus 

 is on the plane at which the phenomenon known as coma is 

 most pronounced, which occurs where there is the greatest dif- 

 ference between the numerical values of these two aberrations. 



While there is just enough similarity between these two 

 sets of diagrams to explain how the geometrically simpler con- 

 ception of Sturm was suggested and enough correspondence 

 of the focal lines of that theory with the mean of the focal 

 values when correctly determined to make the calculations of 

 some practical value, the whole theory should certainly be 

 replaced by one conforming to physical observations. 



