62 GEOMETRICAL RESEARCHES, 



8. If be a point, and A B a straight line : then by " rectare 

 O'A B " we mean J A B sin (0 A B) in which the lines 

 O A, B are regarded as of like signs, and in which account is 

 taken of the formation of the angle A B. 



9. By the " rectare " of any closed n'gon A X A 2 . . . . A W A X we 

 mean the sum of the n products. 



J O A 1 . O ' \ ' sin (0 AJ A 2 ) 

 i OA 2 -OA 3 -sm(0 -A 2 A 3 ) 



I OA n - OAj-siii (0- A^Aj) 



in which is any fixed point in its plane, and in which all the 

 straight lines issuing from are regarded as of like signs 



Ifgp The rectare of a closed n'gon of prescribed formation is 

 constant in sign and magnitude. 



10. If S represents any straight line or plane, and that p 

 represents any point ; then p, S represents the length of the 

 perpendicular from^> on S. 



I think it proper to mention, that early in 1862 I forwarded 

 enunciations of the theorems in papers 2 and 3, and some of 

 those of paper 4 to the President of the Queen's College Galway, 

 and to Professor Chasles. I sent also enunciations of the princi- 

 pal porisms and theorems in papers 1 and 2 to Sir James Cockle, 

 the Chief Justice of Queensland, immediately after his arrival in 

 the Colony. 



