(66.) The equation The equation at a,& + aqx + 2328=yo admits of a direct solution only when a=be. First Method. Assume 62 + a, il + QgcX2 + az x3 = (b +cx)?, and a=2bc; then a,' – 46*a, аз Second Method. Assume b2 + ax + a222 + azx=(b + cx + exo)?, and a=2bc, ag=c? + 2be; Az -2ce then (E. 26, 7.) e? 46° az If a is not a square, one solution must be obtained by trial, let this be atah + a, ha + azh'=k”, then by assuming x=y+h, the equation is reducible to the above form. (B. 184, 5; E. Pt. II, 112~27.) Solution of the equation a + ax + agw? + azx3=yä. [1] Suppose a=63: Assume 63 + Qqw + Qgw? + Q303 = (b +ex), then an 362 [2] Suppose ag=c: a + a,& + 22x2 + c* x=(e +cx)), e3 then and x= assume and x = 3ce-a, [3] Suppose a=b, and ag=c?: : assume then If neither of these conditions is fulfilled, one solution must be obtained ; let this be a + anh + a, ho + azh=k”, then by assuming w=h+y, the equation may be reduced to the form [1] (E. 147-61; B. 1915.) Solution of the equation ax + cy' =x'. Assume x=ap3 – 3cpq', and y=3apřq-cq', then s=ap? +cq. (E. 187.) y = 3t u + 3atu’ + (a’ – b)u”; (B. 196.) FORMS OF BIQUADRATES. (67.) Every even biquadrate *.24.n. Every odd biquadrate # 24. n +1. 04, 14, 2*, &c. (La)*, when a is even, 04, 1o, 2o, įsa-1)]', when a is odd. Remainders of 4th powers from every modulus from 3 to 12. Impossible forms of equations of the fourth degree. ** +y*=**, and generally, ma*+may=x, m being a number of an impossible form to modulus a, and n prime to a. No triangular number >1 is a biquadrate. (Leg. 324-7; B. 71—6.) a1 INDETERMINATE EQUATIONS OF THE FOURTH DEGREE. (68.) Solution of the equation a+a,x+Qx*+azw3 +@qw*=yo. [1] Suppose a=52: assume b + a,x + agu? + a 3.213 + a4w* = (b + ex +fxo), e and f= 2b 25 then az – 2ef f2 — aĄ [2] Suppose an=c?: assume a + ax + aza? + azw3 + c*w*=(f+ex + cw*)', a, e and 2c f – a then a, -2ef [3] Suppose a=b', and a,=c?: assume b? + a,x + QgQ2 +2.3.23 + c*** = (b +ex + cx?)®, a 1 e + 2bc - ag then 26' az F2ce аз 2c e= and x= If neither of these conditions exists, let one solution be found, a + anh + a,h? + azh + agh* =k, from which, and the given equation, h (K* - 8 ak + 4a). 4a? - 364 M If the equation is a + ag X? + 04@* =yo, let one solution be found, a + a,ho + ach'=ko; a,h + 2a,h a + 6a,ho-p. k 2k then if p= and a (qe + 3a) h - 2pq (E. 128–46; B. 186-90.) q– 24 Solution of the equation xo + cy'=x4. Assume x=p* – 6cp?q? +c*q*, y=4pq(p? — cq°); then x=po + cq°. (E. 198.) Solution of the equation 2? + axy +by' =**. Assume x=+* – 66ťuo — 4abtu – (a' — b)bu“, y=4t'u + 6at u+ 4(a’ — b)tu+ (a’ - 26) au; then x=t+atu + bu'. (B. 197.) or (69.) Solution of the homogeneous equation f(x,y)= e; in which w and y, as well as y and b, may be considered prime to each other. Assume x=cy +ex, c being such that ac" + a_0"-1 + a,c" – 2 + &c. + ante; by substituting this value of x, the given equation is reduced to by" + b yn – 1% + ba yu – m2 + &c. + b^x" = +1. There cannot be more than n values of x, between the limits + de and - {e, that render the integral polynomial Q(x). (B. 88.) p and To determine 9, the values of y and >, which render f(y, %) a minimum : let aj, Ag, &c. ar +B. -1, &c. be the roots of the equation bw" + 6,21 – 1 + b2.21 – 2 + &c. + bn = 0, then is the converging fraction nearest to one of the quantities P 9 Q1, A2, &c. a,' &c. which must be determined by experiment. (Leg. 120-8. E. Add. 28.) (70.) Solution of the equation x" — b=ay, a being a prime number, and b prime to a. If n and a-1 have a common factor C, there will be c solutions, or none. If the given equation is possible, b' - 1a. If one solution, x=e, has been obtained, the others may be found by multiplying e by the several roots of the equation xon-1=ay. The proposed equation is always possible, if n is prime to a-1. Solution of the equation 2" – 1=ay. If u is the remainder of ma, all the values of x will be found amongst the remainders of the quantities u, u', u, &c. un-11a, unless two or more of these remainders are equal. If 7 – 1=ay is impossible, r being a root of the equation x" – 1=ay, and m a divisor of n, then r is a primitive root. If ni, ng, &c. are the prime factors of n, the number of primitive roots of the equation 8" – 1=ay will be expressed by ni &c. n. |