338 Mr. A. B. Northcote on the Constitution of Allophane. 



It may be seen without difficulty, that if \', /x' be the values 

 for a polygon of n—l sides, and \", yu," those for a polygon of n 

 sides, then the values for a polygon of n + 1 sides are 



X'" = ytiV'", /^"' = A'X'X" (a/i" - A'/SX") . 



And since in the case of the triangle we have X' = «, fj,' = A'^, 

 and in the case of the quadrilateral we have X" = ^'^, fjj' = cty, 

 where 7=4A« + 2j8©, we can form, step by step, the equation 

 for any polygon. 



From these we can deduce the condition that it should be 

 possible to describe a polygon inscribed in V, and circumscribed 

 about U. These conditions are, — 



For a triangle, /3= ; 

 . . . quadrilateral, 7 = j 



... pentagon, S = 0, S = «'^7— A'/3^; 



... hexagon, e = 0, e = S — A'7^; 



... heptagon, ^ = 0, (f) = a.^ye — h'^ ; 



... octagon, '»|^ = 0, 'v/r = 50— A'/3V; 



and so on. I suppose these values will be found to coincide 

 with those found by a different method by Mr. Cayley in a 

 former Number of this Journal, but I have not verified this. 



Trinity College, Dublin, 

 April 13, 1857. 



XLVII. On the Constitution of AUophane. By Augustus 

 Beauchamp Nortucote, F.C.S., Senior Assistant in the 

 Royal College of Chemistry^. 



THE occurrence of the mineral allophane at New Charlton, 

 near Woolwich, appears to be quite of recent discovery. 

 In the summer of last year, my friend Mr. Bullen, of the School 

 of Mines, observed this substance occupying the position of a 

 vein in a chalk-pit at that place, which from its existing at both 

 sides of the quarry, had in all probability stretched across before 

 the clearing away of the chalk had been effected. This vein had, 

 it appears, about the same time attracted the attention of other 

 observers, for on the 19th of November last Mr. Morris read a 

 paper before the Geological Society upon its existence and com- 

 position f. 



My own analyses of the specimens which Mr. Bullen was kind 

 enough to procure for me were unavoidably delayed ; but not 



* Communicated by the Author. 



t Quart. Journ. Geol. Soc. vol. xiii. p. 13. 



