192 Sir David Brewster on the Changes in the Structure 



after commuting the symbols and differentiating the result i 

 times, we should find 



(^ + ^)D i ' +2 M+(2i + 4).rD ,+1 «-(m-(i + l)(i + 2)) M =D'X. 



Making (i+l)(i-f 2) = »j, this becomes 



(/c 2 + ar 2 )D t+2 a + (2i + 4>D i+1 « = D'X, 



which is integrable when multiplied by (A 2 + a? 9 ) +1 . 



Each of the values of i gives a solution, and the two together 

 the complete one. It may be remarked, that we might integrate 

 i times instead of differentiating, integrating each term of the 

 equation by parts. The solutions will not be practicable when 

 i is not integer, but they may often be transformed by the in- 

 troduction of a definite integral. 



January 28, 1852. 



XXX. On the Development and Extinction of regular doubly re- 

 fracting Structures in the Crystalline t Lenses of Animals after 

 Death. By Sir David Brewster/ K.H., D.C.L., F.R.S., 

 and V.P.R.S. Edin.* 



[With a Plate.] 



SINCE the year 1816, when I communicated to the Royal 

 Society an account of the doubly refracting structures 

 which exist in the crystalline lenses of fishes and other animals, 

 I have examined a great variety of recent lenses, with the view 

 of ascertaining the origin of these structures, the order of their 

 succession in different lenses, and the purpose which they 

 answered in the animal ceconomy. Although I had found that 

 in the lenses of the cod, the salmon, the haddock, the frog-fish, 

 the skate, and several other fishes there were three structures, 

 the innermost of which had negative double refraction, the next 

 ])ositive, and the outermost negative double refraction, yet in the 

 lenses of animals the greatest discrepancies presented themselves. 

 In every case, however, excepting one, I have found the central 

 structure in all quadrupeds f to be positive, while it is always 

 negative in fishes when there are three structures ; but this posi- 

 tive structure sometimes existed alone, with faint traces of a 

 negative structure ; sometimes it was followed by another jwsi- 

 tive structure, separated from the first by a black neutral circle, 

 in which the double refraction disappeared. Sometimes these 

 two positive structures were succeeded by an external negative 

 structure. Sometimes the central and external positive struc- 

 tures were separated by a negative structure, and at other times 



* From Phil. Trans. 1837, p. 253-258. 



t Excepting the hare. See Phil. Trans. 1836, p. 37. 



