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a

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-n

2n

{iminop)

&c.}

an

ma

+

xan

(15.) Development of Sq.com- } (a + bx^)by series.
Si com-1 (a +60%)=
b

62
apa
px

p(p-1).mn
+

+
a mtn 1.2.a?

m +- 2n
63

p(p-1)(p-2) 2.3n +

+ &c.
1.2.3. as

m + 3n
1
px

a?

p(p-1)x - 2n

+ Im+np b'm+n(p-1) 1.2.62' m+n(p-2) 20XP+1

(m, n) ax (m+npb

(m+np) (m+n.p-1)62

(m,n)(m - 2n)a^x – 3n +

. &c. (m + np)(m + n.p-1)(m + n.p-2)68 2 XP+1

(m+n.p +1) 6

m(m + n)a
(m+n.p+1)(m+n.p+2)62

&c.
m (m+n)(m + 2n)a:
1

m+n(p+1) 2011 XP+1

+
(p+1) na (p+1)(p+2) no q: X

(m + 0.0 +1)(m + m.p+2)
t

X2 + &c. (P + 1)(P + 2)(P + 3) noa?

1 OM XP

pna
+

X-1
(m + np (m + np)(m+n.p-1)
p(p-1) n'a

X-?+ &c.
(m + np)...(m+n.p-2)
pnb

p(p-1) n26
+

&c. m(m+n) x m(m + n)(m + 2n) 1

X oom n -XP+1

(p+1) nb (p+1)(p+2) n°62.00

(m, n)(m — 2n) X2 +

&c. (P + 1)(P+2)(P + 3) n°63

(Tr. L. 175-80; Hirsch, Int. Tab.)

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+

&c.)

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m

n

{c+1

EE

b + cy

ez=0;

becomes loce+ezyje +exy?)??

or

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R(x) (16.) Reduction of

(a + a,& + a, x2 + azw3 + aq&") } Assume a=

then the denominator, which may be

1+ y represented by X, becomes

(e +e, y +egy + ezys +exy')}

(1 + y) in which e, e,, &c. are functions of a, a,, &c. b, and c. Let the values of b and c be determined from the equations

en=0, Rw)

R(y)

R(y) then X

Y this again may be separated into

R,(y) y.R,(yo)
S"
+

Y
y.R 9

R(x)
S

becomes
Y

' which may be integrated by preceding methods.

R,(yo)
If R, is an integral function, each term of S, may

Y
be integrated, or reduced by preceding methods to the form

Soor/ If R, (y) is a fraction, it may be decomposed, and each separate fraction may be integrated, or reduced to the form

fua

(+ a) Y
a + ax + a zd
x?

R(Q)
R(w)

may be reduced to the form
b + b/w + 6,82

X by multiplying the numerator and denominator of the fraction by the denominator.

Sizle+exx +exo)]

у? WY:

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X

ume

R(x) S. * (a + a, iv* + a,&* + agx6); may be separated into two parts

R(2) #Spe by separating R(x) into R2(x*), and x. R,(x”), and putting wʻ=y.

R(x)
Site
* (a + a, m? + a, «"); may be separated into two parts
R (x*)

8.R(0%)
face (a +0,24%ax}}and S: Ta +0,7**%!
to reduce the first of these,

(a.+ ay.x2 + &*)&=Xy; to reduce the second, assume

(a + a,m? + 2qw*)&=x; then by substituting for æ its value, we obtain respectively yo.Rz(y)

z?.R_(x4)

and Sota-4a2, + 4 ay)}

(a' — 4 a f_R(w).a + a, v2 + a, x** and SR(a).a + aqx++ a&^%** may be transformed in the same manner as the preceding formula. To reduce SR(x).a + a,x + aqx* + a 3x37** assume a + ax + a? + az 203

aš + xz, or=aztw + %, by either of which assumptions, the term dzw will disappear from the radical.

R(x)
To reduce Su Ta + a, * + ax* + awe

, separate

126 + ax8)* R(x) into R, (x) and X.R2(x), and put y=x?: the part corresponding to R2(x) may be further reduced by assuming

1 y +

Y

(L. C. D. 406—11.)

Rw)

b.b2

+

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R(2) (17.) Integration of

(a + a, x2 + a264)
Let a + a, x? + agw*= (b, +,x*)(b, + cowo),

=b.6,(1 +e2.x)(1 +eg.x?).
[1] Let the quantities ez, e, be very unequal, and e, > ez;
assume e, wo=y’, then

R(Y) Sad

becomes Sy *(a + a,x2 + +)?

ei

y 1+ and since (

1 may beexpanded by the binomial theorem in a series which converges rapidly, if oz is small, the required

ei R(y)

(1 )+ [2] If

ei

and e, are nearly equal, put the given function under the form

R(2) en ez

1+

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e,

integral will depend on So T– ay“).1 + y

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1

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1 = p? - k, and

ea

= pi + k, then by substituting

2

2

these values we obtain /: {(+202)2 – k* }

R(20)

1

1.3

= S.R(w).

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+

1 +

+ &c
(202 + x 2' (r2 + x) 2.4(+ x)
a series which converges rapidly if yoo is positive, and k small :
the required integral is thus made to depend on

R(2)
Son
= (y2 + x*) – a

a b + 0,20914
Lagrange's Method. Assume y=

then

b, b, +0202
R(2)

R,(yo)
(1 **1 ]

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2

2

where p*yand q*y* are either both positive or both negative, and

p=(+0,62)} + { + (0,62 b,c,)}t,

q=(+0,6)}-{+Cbb,c,)}},

or q={ +(czbą b,c,)}}-(+0,ba); + or - according as C, and b, have the same or different signs: and the second value of q must be taken, if c,b, and b,c, have different signs. The same transformation may be repeated by assuming yQ

Pi=p+ (p? -99), 1+q*y

9.=p- (p? 9')}; 9, , Y,

, Pe=P. + (p,' – 9,3)}, 92=P2-(P-9)}; 1+9iyi &c.

&c. where Qm represents {(1 P.y)(1+ny)}: or the values of P1, 91, &c. may be thus calculated ; let P + 9 =?,

P-9 =s, P. +9=19 P1-91=812 &c.

&c. then

r=r+s, and s, =2(rs).
&c.

&c. by this process we obtain a series of the form S,R(9°) +S4R(99) + &c.

R(Y)
Y

(10 in which, since pq continually increases, Px - 9n may be made as large as we please.

Similarly by assuming
p=P-1 + (P4-92)}, q=P-1-(P3-92),
&c.

&c.
P-1=(+9),

9-1=(pq), &c.

&c.

+

or

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