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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

Cm+i+F(cm)
Ex367.P
+ 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)

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and let

Cg

=k, then the integral of (1) is
C1

ei
2*+1 + k.e+

+a.

e* + k.ee If b=a, and c=1,

then Un=tan {(x + const.). tan-'(-a)-1}. The equation wx+1Uz+a..Ux+1+byUr +co=0, may be reduced to

Vi+2+(b: - Q5+1) ve+1 + (x - 27.6,)v, =0,

(1)

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(32.) The equation of the third degree

Ug+2•Ur +1U, - 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=

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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
Un= A2 + B...C+Cx.Pro

if gr=0,
U,=A, +1B..C++CcPx + 'Ex.d.P.x2 if lg.=0;

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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

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+

1.2

1.2.3.4

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+

1.......6

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a

1
1

1
= log.b+

Gera 'am +6

2 (a a +b) 2 (ax + b)2

64.a +

&c. + const.

B
4(ax + 6)*
1

1

1 S.

mbe

+ zm (m - 1).com

m(m + 1)(m+2)64 +

. &c. + const. g 2.3.4200 + 3

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2.00+1

Śm log, m= log, 2x + (x + ?) log, x– x +

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m

am + b

a

1

1 -logb +

+ Sm(-1)" 2(aw+b)

2m(ax + b)2m + const.

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S Šm log, m= £ log 2* + (x + 4) log, « — x + Sm(-1)-1.

2m (2m - 1)x2m-T

62m

Ss

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