Vi-1 Vi-2 being constant, and assumed equal to unity. Let this continued fraction be represented by F(cz), then will Pm{F(cm)} be a particular integral of v, in (2), and *-2 v=Pm + F(cm) *C and RC being arbitrary constants: this value of v, multiplied by Pm (–am) will give the required value of From this the value of u, in the equation Ur+2 + ',• Uz+1+*Qz.U=bx may be obtained: the result is =Pm m{ – 'àm:F(Cm+e)} * Cm+i+F(cm) Cm+2• F(cm) in which a constant must be added after each integration. *--2 Ux' é-C **** + F(-)),{,.. (31.) The equation of the second degree, Uz +1.Uz - auz+1+buz+c=0, , (1) in which the coefficients are constant, may be reduced to Vx+&+(a + b)vx+1 + (ab+c)v=0, (2) and let Cg =k, then the integral of (1) is ei +a. e* + k.ee If b=a, and c=1, then Un=tan {(x + const.). tan-'(-a)-1}. The equation wx+1•Uz+a..Ux+1+byUr +co=0, may be reduced to Vi+2+(b: - Q5+1) ve+1 + (x - 27.6,)v, =0, (1) (32.) The equation of the third degree Ug+2•Ur +1•U, - Q (4x+2 +Uz+1 + 3x)=0, may, by assuming u.= tan vs. Va, be reduced to Vx+2 + 0x+1 + '.x=0, the complete integral of which is Vr=cq.coss T.X+Cg.sin T.8, then Un=va.tan (c.cos ft.X+Cg.sina.«). (Tr. L. App. 386; Mem. Analyt. Soc. 1813. pp. 84-95.) EQUATIONS OF MIXED DIFFERENCES. (33.) Equations of mixed differences, in which Ux, Uz+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 ur=vx+k: the equation Un + aur +1 + bd,u, + cd, Ux+1+f=1, f may by assuming u,=0, be reduced to 1ta Vrtav+1+bdou.+cd,Vx+12 which is satisfied by putting v,=ekt, and ez, ez, &c. being the roots of the equation 1 + aek + bk +cket =0, f U;='c.e;" + c.ee, + &c. 1+ a The equation Uz - (1 + a.€*) d, U, + b(1 + a.€*+1) Uz +1 +C, may be thus reduced to an integrable form : let u,=X,• V.,, in which >, is determined by the differential equation %.-(1 + €*) d.&.=1; from which we obtain %. then the proposed equation 1 + aet? becomes V, + bvc +1 -0.07 +c=0. (34.) The equation d.uz +1 + a,.d.u. +b,.U3+1+c..u,=e, may, by assuming Uz+1 +2.0,= v., be reduced to d.v. + 6.7.V.=e, +(d,a, +27.6.-C.) U., which becomes dx0, +62.V.=en, if d,a, +27.6.-Cr=0: if this latter condition is not satisfied, assume d2Qz+az.bz - Cc=8x9 0,0,+6.0, 1 then Uz= the equation u, +1+Q,.u,=v, becomes d.0;+1 +'az.240, +bx+2.0,+1 + 'cr.0,='en, 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 d, ax + 22.67 -0,=0, when this condition is satisfied, a value of v, may be determined from the equation dov, + bc.v.= e and from that, by successive substitutions, the value of uz. If we begin by assuming dru, +6,-1.4,= v. the above transformations may be effected in an inverse order : the steps of the operation are analogous to the preceding. The values of x obtained by the first method are if gr=0, and those obtained by the second are Uz=A, +BL.C+C2Qx.Pro if gr=0, u='4,+B4.C + 'C:S**QxPx + ter/'R.P.x if 'g;=0; &c. &c. in which p is any periodic function. (TT. L. 887, 8; L. C. D. 1956–63.) SUMMATION OF SERIES. (35.) Let u, + u, + &c. +u, be represented by S. Um then Smum=2,47+1 + const. = EU3+1 – Ex=0Ux+1** 1 (36.) Š,(-1)n-1.9. (22—1) 62 (24-1) 64 =(-1)" {} m(m-1)(m-2).am (20 – 1) 66 m (m— 1)...(m - 4).com-5 + &c. + const. &.} 20" + - 3 + 1.2 1.2.3.4 + 1.......6 a 1 1 Gera 'am +6 2 (a a +b) 2 (ax + b)2 64.a + &c. + const. B 1 1 S. mbe + zm (m - 1).com m(m + 1)(m+2)64 + . &c. + const. g 2.3.4200 + 3 2.00+1 Śm log, m= log, 2x + (x + ?) log, x– x + m am + b a 1 1 -logb + + Sm(-1)" 2(aw+b) 2m(ax + b)2m + const. S Šm log, m= £ log 2* + (x + 4) log, « — x + Sm(-1)-1. 2m (2m - 1)x2m-T 62m Ss |