1003.] 



On Skeiv Refraction through a Lens, etc. 



515 



It commences (at distance 7 '7 2) with contact of the two branches, 

 their common tangent being also a tangent to the primary focal line. 

 The lower branch then passes through the upper, giving three areas 

 separated by two points of self-cutting. These two points then travel 

 outwards, causing the two loops beyond them to become smaller, till, 

 at distance 8*51, the loops have vanished and are replaced by cusps, 

 which are at the two ends of the primary focal line. 



The third row consists of sections intermediate between the two 

 focal lines, the cusps having changed to rounded off angles, and one of 

 the sections (at distance 11) being approximately an equilateral triangle 

 with corners rounded off. The last of them, at distance 13, has a 

 fairly sharp angle at the top. 



The fourth row consists of seven sections through the secondary 

 line and one at a greater distance. The first has a cusp at the top, 

 changing into a small loop, and then we have figures of 8, with the 

 upper loop enlarging and the lower diminishing, till at distance 18 "51 

 the lower loop has been replaced by a cusp. There is thus a cusp at 

 each end of the secondary (as well as of the primary) focal line. 

 When the figures are produced experimentally by receiving the pencil 

 on a card, it is possible to hold the card in a very oblique position 

 in which the two sides of the figure 8 unite, and give a single 

 line resembling the image of a slit. This image is the secondary 

 focal line. 



When the secondary line is passed, the area of the section increases 

 rapidly. The form is at first that of an oval sharpened at the bottom, 

 as shown in the last figure of the Plate ; but, as the distance increases, 

 the two ends become nearly alike ; and at still greater distances, the 

 lower half becomes broader and blunter than the upper, as exemplified 

 in the section at infinite distance which stands first in Plate 10. 



19. In the figures of Plate 9, the zero point x = 0, j; = 0, through 

 which an undeviated ray from the origin would pass, is indicated by a 

 large dot, and it always lies below the centre of figure — a consequence 

 of the preponderating action of the lower portion of the annulus. At 

 distances 6 and 9 '9 the zero point coincides with the 180° point and 

 the 0° point respectively, and at intermediate distances it lies outside 

 the curve. This explains the observed fact that, when two concentric 

 annuli are employed, differing greatly in size, the curve due to the 

 inner often lies outside that due to the outer. In dealing with the 

 solid pencil given by a very obliquely placed lens, it is clearly not 

 permissible "to identify the bounding surface of the pencil with the 

 surface constituted by rays from the outermost zone. 



20. The direction-cosines I', m', n, are approximately simple-harmonic 

 functions of the angular Co-ordinate 6 of the point of emergence. The 

 amplitude of the chief term (of period 360°) is some fifteen or twenty 

 times as great as that of the next term (of period 180°); and when 



