RECIPROCAL EQUATIONS. 



g, — , &c. Then from the theory of equations we may con- 



fider this to be made up of the faAors («■ 



(*--i)(.V-y)(^-.)(-V-^), &C. 



a ' 



^ . 1,1 I 



Or, puttincr a A ■= ni, 6 -{■ ~-- =: n, c + — = J"? 



a b e 



&c. thefe become 



(x'- + m* + l) (^^ + n* +.J) («" + r.r + l), &c. 



If, therefore, we really perform this multiplication, and 



Find the two values of z in the equation 2' + (^ — i\ 

 -+?—/— I = o, and call them r, r' ; then x = — j 



*• -^ i*- ± ^/ (i r' - I), ^ = i ^' + ^/ (I ^'. _ l) 

 6th deg. x' +px'' + q .v» + rx' \- qx' + px- + I =0. 



Find the three values of 2 in the cubic equation z' + 

 Py + (y — 3) 2 + ,• _ 2/> = o, and call them r, r', r"; 

 then 



»= = S*- ± -v' (i '■' - I) .V = i »• + V (i »•" - I) 

 * = ir"+ ^(^r"^_ I) 

 7thdeg..v' + /..v« + 9x> + (-.T'+;rx5 4-«*^ + /.x+i = o 

 Find the three values of z in the cubic equation 2' + 



equate the co-efficients, it is obvious, fince the multiplication (/' ~ i)z'+ (^q— p — z')n.-\-r — p~q +. i = o, and 

 is reduced to half the number of faftors, the equation by =^1' ^heifi r, r', »■"; then x 



which the values of m, n, r, &c. are obtained, will be of 

 only half the dimenfion of the original equation ; and having 

 fouad thefe, fince 



x"- + m X + I =0 

 x' + nx + 1=0 

 x' + r K + 1=0 



we (hall have x = + 



2 ~ 



-Y±^ 



(t - ■)• 



&c. 



Thus, for example, kt there be propofed the equation 



A-' + 5 ^' -f. 7 x' + 5 X + I = O. 



Multiply together x'' + m x + i 

 and x' + n X + 1 



.V = i r + V (5 '•' - I ) i ■■' = I '•' ± ^/ (i r" _ , ) 



x=:^r"± ^{^r<"- I) 



8lh deg. x' + px'' + q x' + ' . . . g x' + p .t, + i = q. 



Find the four values of z in the equation 2^ + / 2> -{- 

 {q - 4) '^^ + (r - S p) ^ + s - 2 {q ~ 1) = o, and 

 call them r, r', r", /•'" ; then 



x= if + ^(Ir- - l);x= iW + ^(ir" - l) 



9th deg. x^+px''+ qx-^ + qx"- +px + 1 = o. 



Find the four values of 2 in the equation 2* + (a — i ) a' 



■^(q— p— $)z^ + {r~q— 2p+2)z+s — r~ 

 q + p + I = o,and call them r, r', /■", r'" ; then x = — i 



= h' 



r 7 ^^ 4- m nl x'^ + m 1 



+ « 



Comparing the co-efficients, we have m + n =2 ^, and 

 mn + 2 =^ 7. 



Hence, m = 



and 



nd n =: 



5 - V5 



± Vdr^ - I); *=ir' + ^(^r'^ - i) 

 ^= \r"± ^/{^rl"- l); .v=ir'"+ ^/(i,-'"^- l) 



A reciprocal equation of the loth and higher powers, 

 requires the general folution of equations of the 5th and 

 higher powers, and therefore cannot be exhibited analyti- 

 caTly. Bonnycaftle's Algebra, vol. i. 



Binomial equations are all reciprocal equations of a pe- 

 culiar kind, which renders them all refolvable by means of 

 certain trigonometrical formulae. 



Binomial equations are all reducible to the form .r" + 1 

 = o; or 0,-"'= I ; or x"'= — I. Where it is obvious, 

 that if »i is even, or »; = 2 «, then x"- " = 1 will have two 

 real roots, -viz. -f- i, and — i ; and x" =l — i wiU have 

 two of its imaginary roots + V — '> and — ^/ — i ; fo 

 that, in both cafes, fuch an equation may be reduced two 

 degrees lower, by dividing it by ^'^ — i, or .r^ + i, and 

 the refulting equation will be a reciprocal one, having 

 each of its co-efficients equal to unity. If m be odd, then 

 From the preceding principles are readily deduced the fo- the equation will necefTarily have one real root, and no 



more, which will be -)- i in the firft cafe, and — i in the 

 fecond ; confequently, fuch an equation can be reduced but 

 one degree, the fame as thofe above dated. We may, there- 

 fore, find a direft folution for all binomial equations of odd 

 dimenfions as far as the 9th power, and of even dimenfions 

 as far as the loth power, by the principles and formulas al- 



± a/ 



30+ 10^/5 _ \ 



-■) 



( .. 



/30 - 10 V 5 



4 ~ ■ V 16 



which are the four roots of the propofed equation. 

 From the preceding principles are readily dedui 

 lution of all reciprocal equations under the loth power, 



3d degree, x^ + px'^ + p x + 1 = 0. 



Find a in the fimple equation 2 — /) — i =0, and call it 



then 



X = — 1, X = ^r + ^ {^1 



- 



4th degree, x' + p x^ + q x' + p x + 1 = 0. 



Find the two values of 2 in the equation z' -j- p % + q — 

 2 = 0, and call them r, r' ; then 



5th deg. x'^+p:if + qx^ + qx'-\-px + I =z o. 



ready given, by merely making A = i, q = i, r = i, &c. 

 and it would, therefore, be ufelefs to repeat them again in 

 this place; we fhall proceed immediately to the general 

 folution of binomial equations, on the principles of analyti. 

 cal trigonometry. 



All the imaginary roots of the equation 



x" — X = o 



3 U 2 are 



