366 PHILOSOPHICAL TRANSACTIONS. [aNNO 1778. 



Others of a like nature may be derived from the general theorems already inves- 

 tigated; for the circle, by returning into itself, often reduces them to a simpli- 

 city to which there is nothing analogous in the hyperbola. Many examples of 

 this might be adduced, but the two following may suffice. (1.) Let abode (fig. 

 6) be a regular polygon inscribed in a circle, and let m be the number of its 

 sides; it is required to find the sum of the lines fa, fb, pc, &c. drawn from any 

 point F in the circumference, to all the angles of the polygon. By the method 

 which in art. 8 was employed to obtain the sum of the sines of a series of arches 

 in arithmetical progression, it will be found, that the sum of the chords of the 

 arches a, a -\- x, a -{- 1x, . . . . (m), that is, (making fa = a, and ab = o^) the 

 sum of the chords of the arches fa, fb, fc, &c. = 

 cho. u— cho. (o 4- w;j:) — cho. («— .r) + cho. (rt 4-H(i — ,c) i ^ • .1 .. - 



t: — -; — ^^ r^ ^^--^- -\ but, m the present case, mx is 



2 X ( 1 — COS. A.17 ' r 5 



equal to the circumference, and therefore — cho. {a + mx) = + cho. a (the 

 chord of an arch greater than the circumference being negative); and, for the 

 same reason, cho. {a -\- mx — x) = — cho. {a — x) = -j- cho. {x — a). Hence 

 the general expression becomes ^ o.a + c-io.{x — a) _ j.,^ _|_ ^^ _j_ p^, _|_ _ ^ _ ^^^ 



If therefore gk be drawn from the centre, bisecting the chord ab in h, and 

 meeting the circumference in k, the sum of the chords, tiiat is, fa + fb -|- fc 



-f FD + FE = X GK. 



FK 



(2.) Let n be an even number, the rest remaining as above, and let it be 

 required to find the sum of the n powers of the chords, that is, the sum of fa" 



+ FB" + FC" (ffl). By reasoning, as in the case of the sines, it will appear 



that, if p be the greatest co-efficient of a binomial raised to the power n; a the 

 co-efficient next less than p; b the co-efficient next less a; and so on; then, 

 (cho. «)" = ;) — 2a cos.a + 2b cos. 2a + 2d cos. 3a + &c. 



(cho. a + x)" = p — 2a cos. (a + x) + 2b cos. 2 x (a + x) + 2d cos. 3 X (a + .r) &c. 

 (cho. o + Sx)" =p — 2a cos. (a + x) -f 2b cos. 2 X (fl + 2.i) + 2d cos. 3 x (a + 2.r) &c. 

 &C. 



Each of these vertical columns is to be continued downward, till the number of 

 terms be equal to m, and therefore the sum of the 2d is mj). The sum of the 

 3d, or of — 2a X (cos. a -f cos. a -\- x -\- cos. a -f 2.i (m), by art. 8, is 



- 2A X COS. a - COS. (a + m.) - cos, (a - .v) + cos, (a + m,-,) ^ Q,^^^,^^ „,^ ^ t^^ 



2 X ( 1 — cos x) ^ 



;- . cos. a — COS. a — cos. (a — x) + cos. (a — x) „ , ... 

 circumlerence — a X ^^ = O. In like man- 



1 — cos. X 



ner do the sums of all the subsequent columns vanish; and therefore, cho. ci -\- 

 cho. a 4- a.' -f- cho. a -\- 2x (m) = mp. But when n is an even number, 



/> = ;- — : X i • X 7- = — T-TTi i — X 2 . K therefore the 



i« — 1 Jh — 2 hi 1. '2.3.4 hn 



radius be put = r, and the expression be made homogeneous, we have fa" -f- 



FR" + FC" {m) — w X ^±^^^^-1 X 2-"r . a. E. i. , ■ 



