VOL. XLIV.] PHILOSOPHICAL TRANSACTIONS. 36l 



Co-tang. of 77a = r^ + x'e + b7'n + c'tW + d^<'n'+e'<»n*, &c. X J; whereN= -. 

 Or = 1 + a'n+ bV+ cV+ dV+ E'N^ &c. X ^r^f; whereN = ^. 



1 _ „'n + b"'n» — n'N' + n"'N^ — re'^N*, &c. rr , « 



Or = T, — T—rr-z — v/ . , ^'" A — r~rz~ X - ; where n = - . 



XIX. Let A =n' a' = ^.n^ 



B = am' — n" b' = j^.n'A' — n" 



c = Bn' — An'" + n" c' =: i.n'B' — h'a + w"' 



D = era' — Bn'" + aw" — n'"' d' = i.nc' — u'^b' + u^'a'— vT" 



&c. &c^ 



Secant of na = 1 + an + bn^ + cn^ + dn* + en^ &c. X m. 



Or = ; — ; — it;—, r-j-; — ^. , „ ■ X m; where n = - m = 5^. 



1 — n N 4- n N* — n'N' + li' n^ &c. rr r" 



w.^^^— — ^^-^^-^— -^^— ^-^^^— — ^— ^— — — — // rrf^ 



Co-secant of-na = 1 + a'n + b'h' + c'n' + d'n* + e'»', &c. x m; where n = ,m = — _ 



rr ntrt ' 



XX. Let c be the chord of an arc (a) of the circumference of a circle, whose 



CC 



diameter is d. Put n = ^. 



mu ujr ,1+»1— ».„i3-f-n3— re ,54-re5 — » 



The chord of na z= nc -\ r— . — -,— an + — — . — • — bn + -^— . - cw 4- 



Z 5 4 5 O 7 



^-— -" . ■ "~" DN, &c. where a, b, e, &c. stand for the respective preceding terms. 



As the preceding theorems are easily deduced from the first, so the follow- 

 ing are most readily seen to be the immediate consequences of these ; and all 

 depending on no other principles than what are generally made use of in com- 

 mon computations. 



XXL Putting s, s, t, t,/,/', for the sine, co-sine, tangent, co-tangent, se- 

 cant, co-secant, of an arc («), and v its versed sine; let q' ^ \; q"=: ^q'; q" = 



\q , q — -5-7 , q — -5-9 , cstc. jn — ^^ . 



Then s = 1 - q'a + q'" n' - ^"n^ -|- ^''''n* - ^"n^ &c. X a. 

 = a- q'Wr-^ + q"'a'r-* - q^a>r^ + q'""a'>r-\ &c. 

 = a — ifj an -f- ^^ BN — ^, CN + ,-5 DN, &c. where a, b, c, &c. 

 stand for the respective preceding terms. 



And i=r — q'd'r-' -f q"'a*r~^ —q^'r—' -f- q'"'a^r-\ &c. 

 = f — q's -\- 9'"n' — g'N^ -I- 9"'n* — ^'-n', &c. X r 

 =.r — -r^ a'^r-^ + il an — 5-^ bn -)- ^ cn, &c. a, b, c, &c. as before. 

 XXIL Also V = q'a'r-' - q'-'a*r-^ + q^'a"/-' - q^'''a^r-\ &c. 



= ht^^r-'— j^AN _ j^ BN- -i-.CN - ,-^5 DN, &C. 



= TT N — n AN — jij BN— — CN, &c. X r. A, B, c, &c. ss before. 



VOL. IX. 3 A 



