360 I'HJLOSOPHICAL TRANSACTIONS. [aNNO 1778. 



we would transfer that conclusion to the circle, it must be in consequence of 

 the principle which has been just now mentioned. The investigation therefore 

 resolves itself ultimately into an argument from analogy; and, after the strictest 

 examination, will be found without any other claim to the evidence of demon- 

 stration. Had the foregoing proposition been proved of the hyperbola only, and 

 afterwards concluded to hold of the circle, merely from the affinity of the curves, 

 its certainty would have been precisely the same as when a proof is made out by 

 the intervention of imaginary symbols. 



8. Though it might readily be concluded, that the same principle on which the 

 foregoing investigation has been found to proceed, extends itself to all those in which 

 imaginary expressions are put to denote real quantities, it may yet be proper to 

 make trial of its application in some other instances. Let ab, ac, ad, ae (tig. 4) 

 be any arches of a circle in arithmetical progression, and let m be their number; it 

 is required to find the sum of the sines bc, ch, &c. of those arches. Let the 

 radius ap = 1, ab = a, and the common difference of the arches, or bc z= x: the 

 sum of the series sin. a + sin. (« + t) + sin. {a -\- 1x) + (m) is to be found. 



Now, because sm. a = — - — -^ — , and sm.a + x= -r— — ; , 



rj / — I 



&c. ; the series sin. a -\- sin. a -\- x -{- sin. a -{• Ix . . , . (/«) = r-^ — : X 1 + 



cx^-i + c2.ry— 1 (m) — "-—^ X I + c-T^-1 + c— 2x^- 1 (m). 



But these series are both geometrical progressions, and the sum of the first is 

 W=l ^ TTTT— ' ^"d of the second, -^-^ X i_,-..^_. - The sum ot 

 the proposed series therefore is 



C-'v'-' ^ 1— C«^^/-' C-■'^/— I 1 —c-mx ^- 1 1 



C'v'^ ' —€" + ""' X y-' — C-xx a/- ' + €"+'"" -"X v'— ' I 1 



l-CX^- i -C-x^-l + I ' '2^-1 



— C-a^-i + c-''-"'XX a/-' + C- "^xx y-' -c-o-'^+^X -/-'. 

 l-CX^-l—C~x^-i -f. 1 ' 



in which expression, if the sines be substituted for their imaginary values, we 

 , sin.« — sin. (<7 + nix) —sin. (a — x) + sin. (a + »/.r — .r) . i ■ / i \ ■ 



have '- ,,^(il,os..) — = ''"• « + ''"'• (« + ^') + 



sin. {a + Ix) (m). a. e. i. 



When AB = BC, or a = x, the proposed series becomes sin. x -\- sin. 2x + 

 , N, 1 -. , sin. X — sin. (m + 1) x x + sin. mx 

 Sin. 3x- (m), and its value = ^-^J^^^ • 



In like manner it will be found, that the sum of the cosines of the same 



arches, or cos. a + cos. (a + x) + cos. (« + '2x) -\- (ot) = 



cos.a - COS. (. + n,x)- cos. (. - .r) + cc.s (. + nu- - x) ^^^^ ^^^^^ ^ ^ ^^^_ ^. 



2 X ( I — COS. x) ' ' 



, „ . > cos.7».r - COS. (m + \) x x 



'Ix 4- cos. 3x (m) = ; 4. 



^ ^ ' 2 X ( 1 - COS. .r) ^ 



Q. To solve the same problem, in the case of the hyperbola, \vc must follow 



