SINES. 



Of Multiple Arct. 

 By referring to our formula (I), we have 



fin. {a -V i) = fin. a . cof. b + fin. b . cof. a; 



which, by writing (n — l ) a for a, and a for b, becomes 



fin. n a = fin. (n — i) a • cof. a + cof. (n — \) a . fin. a. 



By fubftituting in this formula for fin. (n — i)fl, and cof. (n — i)a, under the form 



fin. ^ (n — 2) a + a [ and cof. j (n — i ) a + a [ , we obtain 



fin. n a = •< fin. (n — 2) . cof. a + cof. (n — 2) a . fin. a {■ cof. a 



+ 4 cof. (n — 2) a . cof. a — fin. (n — 2) a . fin. a [• fin. a 



Subftituting here again for (n — 2) a, under the form (n — 3) a + a, and obferving that the fecond term of the firft 

 line is the fame as the firft of the fecond, we have 



fin. n a — X fin. (n — 3) a . cof. a + cof, {n — 3) a . fin. a >■ cof.' a 



-(- 2 -J cof. (n — 3) a . cof. a — fin. (n — 3) a . fin. a [ cof. a . fin. a 

 — j fin. (n — 3) a . cof. a + cof. (« — 3) a . fin. a [ fin.' a 



= fin. (n — 3) a . cof.' a + 3 cof. {n — i) a . cof.' a . fin. a — 3 fin. (n — 3) a cof. a . fin.' a — cof. (n — ^) a . fin.^ a ; 



the laft term of each of thofe lines being ftill the fame as the firft term of the following ; and therefore, without purfuiijg 

 the operation any farther, the law of continuation is obvious ; -viz. the feveral coefficients will be exadlly thofe of the 

 binomial ; but the figns will be two plus and tvio minus alternately, whence generally 



fin. na = fin. {n — m) a . cof." a + A cof. (n — m) a . cof."'^ ' a fin. a — B fin. {n — m) a . cof.'" ~' a fin.' a — C 



cof (n — m) acoi."'"' a . fin.^ a -j- &c. 



where A, B, C, &c. are the coefficients of (i + i)'". 



And exaftly in the fame manner, we find 



cof. na = cof. {n — m) a. cof." a — A fin. {n — m) a cof." - ' a fin. a — B cof. (n — m) a . cof."- ' a . fin.' a + C fin. 



(n — m) a . cof." ^3 ^ , f,t,.i ^ ^ ^^^ 



In thefe formulae m is indeterminate, and may be afl'umed at pleafure ; let us therefore take m = n, and we have 



fin. na = A cof.""' a . fin. a — Ccof."-'a. fin.' a -)- E cof."-^ a .fin.' a — &c. (II.) 

 and 



cof. «a = cof." a — B cof." 'a . fin.' a + D cof." ^ a . fin.^ a — &c. (HI.) 



where A, B, C, &c. reprcfent the coefficients of (i -f i)". 



From thefe two general formulae we readily deduce the following particular cafes ; viz. 



1. fin. a = I fin a 



2. fin. 2 a = 2 cof. a . fin. a 



3. fin. 3 a = 3 cof.' a . fin. a — fin.' a 



4. fin. 4 a = 4 cof.3 a . fin. a — 4 cof. a fin.' a 



5. fin. ^ a — S cof.'' a . fin. a — 10 cof.' a fin.' a + fin.' a 

 &c. = &c. 



1. cof. a z: cof. a 



2. cof. 2 a = cof.' a — fin.' a 



3. cof. 3 a = cof.3 (2—3 cof. a . fin.^ a 



4. cof. 4 a = cof.* a -. 6 cof.' a . fin.' a + fin.* a 



5. cof. 5 a =: cof.' a — 10 cof.J « . fin.^ " + 5 cof. a . fin.* a 

 &c. = &c. 



formulae" " °^'"°^'' "^^ ""^ continued at pleafure, the law of the coefficients being exhibited in the above general 



In 



