RECIPROCAL EQUATIONS. 



are contained in the general formula 



r 2 kTT 



2 col. .T + I = o ; 



tion x' + 1 = o, or A-" = — i, will have for its generii 

 faftor 



. (2 i + i) T 



x' — 2 col. i — X + I = o; 



n 



i being any integer not divifible by n, and t reprefcutiiig the 



femi-circumferenc*;. For it is a known trigonometrical pro- which, by fubllituting for /■ as above, becomes 



perty, that if , ^ 



1 I .r' — 2 cof. X + 1 = 



2 cof. y= X + — , then 2 cof. ny = x" + -^i n 



r 3 '^ 



2 COi. ~ ;c 4- I = O 



n 



X - 



r 5 '^ 



2 col. — .r + 1=0 



n 



&c. 



from which two equations we readily draw the two fol- 

 lowing ; viz. 



s' — 2 cof. ^ . ^ -t- 1 = O 



x'" — 2 cof. n y . x" + I := O 



•which have neceffarily one common root, being both derived &c. &c. &c. 



from the fame value of .%■ ; and fince thefe are both recipro- , ■ , r , . • 11 .i • . r t . • 



iroiu LUC la , r which formulae contam all the imagmary roots of the bmo- 



cal equations, if x be one root, — will be another ; they have mial equation x" + i = o. 



^ X Suppofe, for example, all the imaginary roots of the bino- 



therefore two roots common, and confequently, from the mial equation a-' — I = o were required, 

 known theory of equations, the former is a divifor of the Here we (liould have 



2 k -x 

 n 



fatter. If, now, we make j = • , or ny = 2 i v, thefe 



equations become 



2 cof. 



360° 



+ 1=0 



, 2ir 

 K^ — 2 col. .V + I = O 



n 



x'"' — 2 cof. 2 i T x" + 1 = 



But the cof. 2 i ff = I, 2 T repr; Tenting the whole circum- 

 ference ; therefore the latter equation is the fame as 



x" — 2 s" -f I = o, or (.v" — i)- = o, 



having ftill for its divifor 



, 2 ,f- T 



X — 2 col. A- -)- 1 =: o ; 



n 



that is, the roots of the equation 



(.r'' — I )' = o, or ;r" — I = o, 



are all contained in the general formula 



, ^ 2 hv 



a' — 2 col. X -{■ 1 = 0; 



n 



and, therefore, by giving to h the fucceflive values I, 2, 

 3 . . . . 5 (?! — i), the following formula; will be ob- 

 tained ; ijiz. 



r 2 . 360° 



x' — 2 col. ~ .X -f 1=0 



n 



r 3 • 360° 



X- — 2 col. j; + I = o 



cof.i:^" 



II 



X -\- I —O 



r 5 • 360 



.V- — 2 coi. - — X -V 1=0 



II 



r 360° / -, 360° >w 



whence, x = col. -^-7- + -\/ I col.^ — — i J 



II 



^= c°f- ^ ± ^' {^°^- — ,- - 



X = cof. &c. &c. 



And if the roots of x" ^ 1 =: o were required, we Ihould 

 have from the fecond general formula 



— 2 cof. 



180^ 



X + 1 = 



2 cof. 



• .V -1- I 



.t' — 2 cof. -? x -f I = O 



>t 



r 6t 

 .r' — 2 col. X + I r= O 



*' — 2 cof. — a; -f I i: O 



r 3-I«0 

 X— 2 cof. X + I = C 



II 



7 - 5. 180 



x^ — 2 cof. :- X + 1=0 



1 1 



&c. 



&c. 



Whence, 



± ^/ 



II ~ V II y 



which contain among them all the imaginary roots of the 



equation k" — i = o. ^ — <^"f- ^^- &c- 



. , .p . , 2 i r For more on this fubjeft, fee Barlow's Theory of Num- 



And it, inltead of making^ = — -, as above, we bg^s, Bonnycaftle's Algebra, and our article Polygon. 



(2i + 1) ■z KsciPROCAL Figures, in Geometry, are fuch as have the 



make_jr = , our fecond formula becomes antecedents and coiifequents of the fame ratio in both figures. 



See Plate XII. Geom.Jig. 2. Here 



A : B :: C : D, or, 

 12:4:: 9; 3 



That 



;<:"■+ 2 j:" + 1=0, or (.v" -f 1 )' = o ; 

 becaufe cof. (2 i k + ■jr) z= — 1. Confequently, the equa- 



