418 
Proceedings of the Royal Society 
equating of such equivalent forms interesting identities arise, and 
this is especially true of the case we have considered where the 
Determinants take the form of alternants. 
12. It has only to he added, in conclusion, that there seems to 
he no easy way of simplifying a Permanent so as to find its value 
for particular values of the elements. The second alternative defi- 
nition given in § 3 is, however, sometimes useful for this purpose, 
viz., when it is possible to find the coefficient of xyz in the product 
by other means. A most curious instance is that of the Permanent 
2 
2 
2 
cos 
cos 
2tt 
2 n + 1 
2.2tt 
2 n + 1 
cos 
n. 2 tt 
2 n+ 1 
2 
2 
2 
cos 
cos 
cos 
2. 27 r 
2 ft + 1 
4. 2 t r 
2 ft + 1 
2.n27r 
2n + 1 
. . 2 cos 
. . 2 cos 
. . 2 cos 
n2ir 
2 n + 1 
2 n. 2i t 
2 n + 1 
n 2 . 2tt 
2 ft + 1 
or 
cos 
1.1.27 r 
2ft + 1 ’ 
2 
cos 
2.2.2tt 
2ft + 1 ’ 
. . . , 2 cos 
ft. ft. 27T 
2 ft + 1 
+ 
which in this way we know must be equal to - 1 for odd values of n. 
5. Optical Notes. By Professor Tait. 
1. On a Singular Phenomena produced by some old Window-Panes. 
The second figure in Professor Everett’s note ( ante , p. 360) has 
reminded me of my explanation of a phenomenon which I have 
repeatedly seen for more than twenty years in the College. When 
sunlight enters my apparatus-room through a vertical chink between 
the edge of the blind and the window-frame, the line of light 
formed on the wall or floor shows a well-marked hink. Similar 
phenomena, though not usually so well marked, are often seen in 
old houses, when the sun shines through the chinks of a Venetian 
blind. They are obviously due to inequalities (bull’s-eyes) in the 
glass which was used more than a generation ago for window-panes. 
Professor Everett’s figure, which was drawn for a cylindrical lens, 
