and Loci of Apollonius, ^c. 89 



The triangles BCI, ICA being isosceles, we have the angle 

 lA or IP right to B = CB right to O ; but PA or PI right 

 B = OC right to B ; therefore the angle BC right to O = 

 angle BI right to P, and the equiangular triangles OCB, PIB 

 give BC : BI : : BO : BP. 



And since the angle BC right to O = BI right to P, we 

 have the angle BC right to I = BO right to P ; and, there- 

 fore, since BC : BI : : BO : BP, the triangles BCI, BOP, 

 are similar, and give OB : OP : : CB : CI. 



Hence, as CB = CI we have OB = OP 



COROLLARY. 



If P' be the position of P infinitely near to or coincident 

 with B, then P'B is tangent at B ; and putting MM for this 

 tangent, we have angle BA right to M == PA right to B = 

 OC right to B, and .-. BM is perpendicular to BO. 



Art. II. — A Communication from the Reverend W. B. 

 Clarke, of Sydney, to His Excellency Sir Henry Barkly, 

 K.C.B., &c., &c.. President of the Royal Society of Vic- 

 toria, on Professor McCoy^s ''New Taniopte^'is" from the 

 Coal-bearing Rocks of the Cape Paterson District in 2Jarti- 

 cidar, and on the Evidence bearing on the Question of the 

 Age of Australian Coal Beds in general. — Communicated to 

 the Society by His Excellency The President. 



[Read before the Eoyal Society, 25th June, I860.] 



St. Leonardos, 1st June, 1860. 

 My Dear Sir, 



I received with much thankfulness the kind reply which 

 you were good enough to send to the letter which I did 

 myself the honour of addressing to youi* Excellency. 



The information contained in it respecting the discovery of 

 Tceniopteris, at Cape Paterson, is very interesting. But I 



