360 PHILOSOPHICAL TRANSACTIONS. ^ANNO 1747. 



and A = t' 



B = AT — T " 



c = bt"— at'"-!- T" 



D = CT — BT + AT — T 



E = Dt" — Ct'" + BT"' — AT '" + t'*. Put K = — . 



&c. ' &c. 



Then the tangent of « + (3 + y + ^, &c. = a +br + CR^ + dk^ + er"; &c. 



XVI. Hence, the sine, tangent, and secant, of any arc a, being represented 

 by s, t,f, the co-sine, co-tangent, and co-secant, by s, i,J^; those of the arc nq. 

 are expressed as in the following theorems. 



Putting n ■=. n. —z-''> n =n •— r— ; n'^ = n . -— ; n' = n .— — ; &c. 

 Sine of na = «a — n"AP + n''BP— w^'cp + n''"DP, &c. X ' ; 



where p = -r-r; a =: *; b = ap; c = bp; d = cp; &c. 



ss 



_. B— 1 « — 2 , n — 3»— 4 n — 5 n —6 . ' 



Or = ns- -2—— - AP -h -^-^-BP - -g-.-^cp, &c.— --> 



where a, b, c, &c. stand for the respective preceding terms. 



l + nl — n 3-|.n3 — n ,5H 



where a := - ; a, b, c, &c. stand as before. 



_, ,1 + nl — n ,3 + n3 — n ,5+n5— n , 7 ^ nj — n - 



XVII. Co-sine of na = 1 - riv + /rp^- n'p^+ n''"p^&c. X -j^, wherep = j.. 



_ ,04-nO-n ,2 + a2 — « 4^.n4 — n . 6-{- n 6-n 



Or = r-|--^.-^Aa-|--^-.^-Ba-f^-.-^ca+-^. -^Da,&c. 

 where a = - ; and a, b, c, &c. stand for the respective preceding terms. 



^ 2»> rr 



OrputM= — X '■»■ N = — — ; A = ^;B = AN;C=BN;D=CN,&C.;p=njp =n— l;p =n— 2,&c. 



And «' = />; 6= ;).|p"'} c-= y.^p^'.^p' ; rf-= p. iy'4p'".ip'";e-= p 4P'"-iP"'-iP""-iP"i &c. 

 The co-sine of na = a — Ba + ci' — dc' -f- Erf, &c. X m. 

 XVIII. Let A = — ra* -}- raw' a'= ^ .m^^ —n ' 



B = + n'" — nn"' + An' b'= j .wV — w"^ 4-n''' 



c = -I- n" + n«' + Bw' — An*' c'= j.^' — ra'V -f- w"' — w'' 



D =: 4.ra''*'_„n''"-|- en- Bn^+An" D'=i.n"c'- n''B'— w''V-n''''-|- w''" 

 &c. &c. 



The tangent of na=.nt-^ A^r—' -j- Bt'r—* -f- ct^r—^ + D<"r-* &c. 



Or = n -j- AN -I- BN^ -f-cN* -f DN* -|- &c. X /, where n = - • 



Or s= no' + A*' + Bc'+ C(i' + dc', &c. where a' = <;6'= Na'}e'= »b';d' = Ke'j &c. 



_ « - >»"n + f»"w^ - »''n^ + «•"'?!«, &c. 

 ^* "" 1-»'h + »'"n' -«'»' + n"'N*.&c. ^ 



