On the Porism of the In-and-circumscribed Polygon. 339 



The texture of these tubes is compact, their colour brownish- 

 black, with a yellowish-browu trauslucency in tl!in layers. Ana- 

 lysis shows them to consist, like the coprolites, principally of 

 phosphate of lime. One hundred parts gave — 



Phosphate of lime 67*53 



Carbonate of lime 4'35 



Magnesia 1-65 



Protoxide of iron 2-95 



Insoluble siliceous sand .... 21 '10 

 Volatile animal matter .... 2'] 5 



99-73 



The microscopic examination of a section shows that the walls 

 of the tube are homogeneous, unlike the coprolites, and that the 

 siliceous sand in the analysis came from the sandstone which 

 incrusted the rough interior of the fossil. The phosphate is 

 finely granular, and retains no vestige of organic structure. The 

 chemical composition and the remarkable shape of the specimens, 

 however, leave little doubt of their osseous nature, unless we 

 suppose them to be the remains of some hitherto unknown in- 

 vertebrate animal, whose skeleton, like those of Lingula, Orbicula 

 and Conularia, consisted of phosphate of lime, a composition 

 hitherto supposed to be peculiar to vertebrate skeletons. 



Montreal, Jau. 5, 1854. 



LI. Developments on the Porism of the In-and-circumscribed 

 Polygon. By A. Cayley, Esq.^ 



I PROPOSE to develope some particular cases of the theo- 

 rems given in my paper, " Correction of two Theorems rela- 

 ting to the Porism of the in-and-circumscribed Polygon '' (Phil. 

 Mag. Nov. 1853). The two theorems are as follows. 



Theorem. The condition that there may be inscribed in the 

 conic U = an infinity of ?i-gons circumscribed about the conic 

 V = 0, depends upon the development in ascending powers of f 

 of the square root of the discriminant of |U -h V j viz. if this 

 square root be 



A + B|-fC^2 + Dp + Ef-»-Ff-fGf +Hf-f- .., 

 then for n = 3, 5, 7, &c. respectively, the conditions are 



C|=0, 



= 0, &c. ; 



