304 NEW SERIES for the 



35. From the nature of the hyperbola, we have 



ab S — ab 2 - S -f- ord 2 - S, ord S 3= 2 ab - S ord - S : 

 22 22 



therefore, by divifion, 



abS abiS , 1 ord i S 



? 



ord S 2 ord ~ S 2 ab ■* S ' 



that is, s 2C -^ -|- - tan - S. 



tan S 2 tan fS 2 2 



This laft formula exprefles a property of the hyperbola per- 

 fectly analogous to that of the circle (Art. 5.), from which we 

 have deduced our firfl four feries for the rectification of an 



arch. Therefore fimilarly, putting s, -s f -s, &c. fuccefiively 



inftead of S, and multiplying by the feries of numbers 1, -, -, 



2 4 



&c. we have as in that article 



1 1 , 1 i 



_ — r - tan - s, 



tan s 2 tan ^ s 



1 



1 , r 1 



= p- + - tan - s, 



2 tan - s 4 tan \ s 4 4 



1 1 , 1 " 1 



4 tan £ j- 8 tan \ s ^ 8 8 ' 



&c. 



This feries of equations being fuppofed continued until their 

 number be n, by proceeding exactly as in Art. 5. when treating 

 of the circle, we obtain 



2 n tan — 



2* 



