Vi-1 Vi-2 being constant, and assumed equal to unity. Let this continued fraction be represented by F(c), then will m{F(cm)} be a particular integral of vin (2), and v ̧ =Ïm { F(Cm+1) } · {1C + °C. Σ2Pm (√ Vx C2 Cm+1+F(cm 1C and C being arbitrary constants: this value of v, multiplied *-2 by Pm (-1am) will give the required value of u the value of u in the equation From this 21 in which a constant must be added after each integration. (31.) The equation of the second degree, Ux + 1 • Ux — α U x+1+bux+c=0, - in which the coefficients are constant, may be reduced to V x + 2 + (a + b) v x+1+(ab+c) v2 =0, (1) (2) +a; let the complete integral of (2) be If b=a, and c=1, then =tan {(x + const.). tan-1(-a) -1}. cx=0, The equation U x + 1 · U x + а x · Ux + 1 + bx • U x + C2 =0, may be reduced to v x + 2 + (b z − a x + 1) x + 1+ (c2 — a ̧·b1)v=0, (1) ·a: let 1v, v, be the two particular x+1 + k.2vx+1 1v2+k. 2vr (32.) The equation of the third degree may, by assuming utan v. ✔a, be reduced to V x + 2 + V x + 1+1=0, the complete integral of which is then uva.tan (c.cos.+c. sinπ.α). (Tr. L. App. 386; Mem. Analyt. Soc. 1813. pp. 84-95.) EQUATIONS OF MIXED DIFFERENCES. (33.) Equations of mixed differences, in which u ux+1, &c. and their differential coefficients do not exceed the first degree, and in which the coefficients are constant, may be rendered integrable by assuming u=v2+k: the equation ux+aux + 1+bd ̧u2+cd ̧u1+1+f=0, which is satisfied by putting v=e**, and e1, e,, &c. being the roots of the equation 1+ae+bk+cke*=0, u2 =1c.e*+*c.e* + &c. The equation f 1+ a u ̧-(1+a.e1) d ̧u,+b(1+a.e+1) U2+1+C, = may be thus reduced to an integrable form: let u,,.,, in which is determined by the differential equation, may, by assuming u2+1+a.u,v,, be reduced to dzv2+b ̧ · v«=e ̧+ (d ̧à ̧+a ̧·b ̧ — c2) ux which becomes d ̧v2+b ̧•v ̧=e, if da+a.b2—c„=0: if this latter condition is not satisfied, assume the equation u2+1+a ̧.u ̧=v, becomes d ̧o ̧+1+1à ̧·d ̧v ̧+b ̧ + 1· V ̧ + 1 + '1c ̧· v ̧ =1e ̧‚ which is of the same form as the original equation, and with which the same transformation may be repeated, until we arrive at an equation in which when this condition is satisfied, a value of v may be determined from the equation d ̧v2+b.ve2, I I = and from that, by successive substitutions, the value of u If we begin by assuming 2-1 the above transformations may be effected in an inverse order : the steps of the operation are analogous to the preceding. The values of a obtained by the first method are (35.) Let u,+u2+&c. + u ̧ be represented by Śmum, then Smum=Σ,u,+1+const. = EU2+1-Σ-02+1" x=0 2 |