83 

 Consequently 



asin.CVR=r's\n. VRC =^ — sin. CRP=.-^ s\n.RPC 



m tn 



= -^ sin- OPC. 



m 



am 



Hence sin. OPC =-,;;: %\r\. $. (i) 



This equation is evidently true, whatever be the cumber 

 of points of refraction between P and F, and therefore is true 

 when FRP is a continued curve as in atmospherical refrac- 

 tion. 



The refraction H, that takes place between P and V = 

 the inclination of the lines P T and R V. Hence 



By equation (1) OC = j- sin. 0. 



The refractive force of air is as its density, and the refrac- 

 tive force in TP is also as >f * — 1, (vid. Newton's Optics, book 

 3, Prop. 10. Horsley's edition, vol. 4, p. 171.) 



Therefore let 6 ^ = A;* — i, h being a constant quantity 

 Then k=VT±Tf and m == ^/ T+T(f) 



Hence OC=a sin. 6 ^ '+^(>> 



^^ l+bf — 'LL sin. d 1 + 4 (p)) 



and OP =r ' , ^ —-^ 



Therefore Ji' = g- 



2{l + fi,) r v/l + *f- — 8in.«9(l+6(f)) _ (2) 



N 3 



