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AN ESSAY
ALGEBRAIC DEVELOPMENT,
THE PRINCIPAL EXPANSIONS
IN COMMON ALGEBRA, IN THE DIFFERENTIAL
AND INTEGRAL CALCULUS,
AND IN
THE CALCULUS OF FINITE DIFFERENCES;
THE GENERAL TERM
BEING IN EACH CASE IMMEDIATELY OBTAINED
BY MEANS OF
A NEW AND COMPREHENSIVE NOTATION.
By THE REV. THOMAS JARRETT, M.A.
FELLOW OF CATHARINE HALL, AND PROFESSOR OF ARABIC
IN THE UNIVERSITY OF CAMBRIDGE.
CAMBRIDGE:
PRINTED BY J. SMITH, PRINTER TO THE UNIVERSITY :
FOR J. & J. J. DEIGHTONS, CAMBRIDGE;
AND RIVINGTONS, LONDON.
M.DCCC.XX XI.
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PREFACHE.
Tux following pages are intended to illustrate and apply
a system of Algebraic Notation submitted to the Cambridge
Philosophical Society in the year 1827, and published in the
third Volume of their Transactions. In that paper the ap-
plications were necessarily few, and the whole was deficient
in that development which was indispensable to render the
introduction of the system into general use at all probable;
but in the present Work it is applied to the demonstration
of the most important series in pure Analysis. The methods
by which these are demonstrated are partly original, and partly
taken from one or other of the works of which a list follows
this Preface, but they are in general so much modified that
a distinct reference to the inventor of each demonstration
appeared useless; so much, however, is due to the admirable
works of Schweins, that it would be unjust not to make a
distinct acknowledgment of the great use that has been made
of his. “ Analysis.” The demonstration of the legitimacy of
the separation of the symbols of operation and quantity, with
certain limitations, belongs to Servois, and will be found in
the ‘Annales des Mathématiques;” and the proof, that the
coefficients of the binomial, (the index being a positive integer,)
are integers, is due to Mr. Miller, of St. John’s College, and
is the only independent proof with which T am acquainted.
iv PREFACE.
The following apparent innovations in the ordinary notation
are not original :
(1) £,¢(«), for p(#+Dz), is partly due to Arbogast,
who uses E(x) for the same function.
n
ae
(2)" d;-76, aor ae is due to Lacroix, although not used
a
by him, being merely pointed out in a single line*;
it was suggested to the writer of these pages by the
analogous integral notation invented by Professor
Airy.
(3) (%),_,> for the value assumed by w when 2 is put
equal to a, belongs to Schweins.
In order that the work may be as independent as possible,
the Reader is supposed to be acquainted only with the first
rules of Algebra, and the fundamental theorems of Trigo-
nometry; and, for the sake of facility of reference, the whole of
the theorems have been arranged in an index at the end of the
volume,
The additions contain a few Theorems of importance that
did not suggest themselves till too late to be, inserted in the
text, together with a few simplifications of the demonstrations
inserted in the body of the Work.
In conclusion, the Author has to acknowledge the great
liberality of the Syndics of the University Press, in defraying
a considerable part of the expense of publishing.
* Calcul Diff’ Tome 11, page 527.
LIST OF WORKS
WHICH HAVE BEEN CONSULTED.
ANNALES des Mathématiques.
Arbogast, Calcul des Dérivations.
Herschel, Examples on Finite Differences.
Hindenburg, Sammlung combinatorisch-analytischer Abhandlungen.
Lacroix, Calcul Différentiel et Intégral.
Laplace, Théorie Analytique des Probabilités.
Schweins, Analysis.
Theorie der Differenzen und Differentiale.
Wronski, Introduction a la Philosophie des Mathématiques.
CHAPTER
i.
Il.
INDEX TO THE CHAPTERS.
PAGE
On Series in general........2.. se seee cere cere eee r cee eee 1
On Products and Factorials...... tiaeashic ete 25. eae ee Mae
On Combinations and Arrangements..........--+-+++++- 20
On Binomials and Exponentials...............--.+-+-+++ 23
On (Finite (Differences cri seteee rein coterie ore rete 40
On Differentiation in general......-....--.-+--++ 20-00 66
On Polynomials eye rier leek rie ei ai aia oe 78
On the Differentiation of Exponential and Circular
WR aACELONS es fies class lee e Oe Tei ome rere ote siete 93
On the Expansion of Circular Functions........---.--- 102
On the Integration of certain Definite Functions........ 123
On Generating Functions... ........--se2 esse eee eee ees 136.
INDEX TO THE SYMBOLS.
PAGE ART.
n
n
1 Zz =, On
nr
g 3 Sin Om
r §
@)
5 15 Seas
n nsr
13 28 Pam, and P,, ap.
15 38 |@ : | a, and | @.
nym n_
n
19 49 Sin tb, § 025 4C-
m m+1l n+l
m,n Mm,ns:s
20 53 C,a,, and C, a,.
m,m—n
21 59 C,,: (a,-,).
n
22 62 Aah.
Al 116 (p+), u-
n
nT Vprty)fu
r r+1
45 125 (uw), and @,_,(w).
126 E, (wv).
127 D,(u).
129A, (u).
58 168 E,,,(u), and D,,, (uw).
66 188 d,"u.
67 195 fru.
78 /9ee S.C):
79 228 wah (a), wa" p(a), wa", and w”a,”.
89 241 (aon Le
nr
136 335 Be.
336 Gy.Uz-
ERRATA.
The Reader is requested to correct the following Errata before he
proceeds to the perusal of the Work.
PAGE LINE ERRATUM. CORRECTION.
5 8 A5+d9n,_1 As... Gena
s
6 last S, Ss,
19 1 cl : c
n+l 1
6 P 1p
nym m,n
21
n—m+1,n n—m-+1,m
22 17 y
last but one +4, a, + a4, 0,4) a,
last 2a,a,” 34, 4,7
m m—l
24 19 A
AT last D,(2x) D,(u)
48 3 $(u) $ (x)
14 a.D. a.Dz.
62 ll a, at
88 ll e2m+s ae2m—S
89 10 2ms 2m
98 6 and last (201) (200)
104 15 ~ n
105 of oe ae
“ last but one Sn Sn
126 8 and 9 x ae
135 6 Sm ae
146 last but one AY (ue) A” (uz)
ON THE DEVELOPMENT
OF
ALGEBRAIC FUNCTIONS.
CHAPTER I.
ON SERIES IN GENERAL.
1. Iw the expansion of Algebraic Functions it has been
usual to investigate the first three or four terms, and from
these to deduce the remainder of the series by analogy. The
unsatisfactory nature of this method in all cases, and the errors
into which it may readily lead us in very many instances, must
have been obvious to all who have made use of it. In some
cases indeed, the connection between the consecutive terms at
the commencement of the series is so obscure, that the most
patient of analysts have given up the search, and have been
compelled to state that ‘the law of the series is not obvious.”
In order to avoid this obscurity and embarrassment, we shall
adopt a notation by means of which the general term will be
obtained in eyery case, and which will enable us to perform
any operation whatever on a series, with the same facility as on
a single term.
2. The m™ term of a series being usually some function
of m, we shall denote it by a,3; and, taking the letter S as an
- n
abridgment of the word Swm, the symbol §,,a,, will be used to
A
2
denote the sum of n terms of which the m” is a,,: that is,
n
Sa Om = A, + Ay + Az + eee + Ay
In this notation it will be seen that the index placed over
the § denotes the mwmber of terms, and that the index placed
under the same letter, is that to which the successive values
1, 2, 3, ..., , must be given, in the function a,,, in order to
form the consecutive terms: that is,
S,p(m) = (1) + O2) + (3) + «+ @(n).
Or, to give examples of a more simple nature,
Se = 1749743? 4+...4+7.
Set 4e4 oe WAR
S,, (r—2m+1)'=(r-1)° +(7-3) + (7-5) +... + (7-2 41).
3. The symbol S, a,, denotes that the 7" term must be
omitted.
4. Theorem. If a, =b,, ("=!)*, then S, Oy = 8.8
m=n
For, since a, = b,
105
az = b,
&e. = &e
ae =.0..
*, A) +d. + a3 + &. + a, = b, + b, +b, + &e. + b,,
n
n
A a
or Sn Gn — Sn bn 7
* By this notation is meant, that this equation is to hold for every integral value of
m, from 1 to n
3
n n n
5. Theorem. Sin (Gn oc bn) 7s Sn Qn + Se Bn,-
nn
For, Sy (Gn + On) = +O, + a +b, +...+4,+6,
=, + d+... +04,+6,+b,+... +6,
= Sr An + ro me
6. Theorem. If b is independent of m, then §,,a,,b=b. S. Gis
na
For, S,,@,0= 46+ a,b + a,b +... +a,b
=) (a, + dy + 34+... +4)
n
=b.Sn4n-
peor. (S,0— 720.
8. Problem. To invert the order of the terms of a given
series.
n
Now, Si Gm = % + Ao + 3+... +A,_1 +4,
= Uy, + Oy_) + Up_o +++. +, by inverting the series ;
n
aa Da An—m41°
In order therefore to invert the series, we must substitute
nm —m +1 for m in the expression for @,,.
n—r
n r
9. Theorem. Si an = Si an + Sh Brim:
For, Sy Gm = $4) + dy + Oz + 00. Gb + fOr + Gyo + --- + a,h
n—?r
re
z =
= Sn Gn + os Dy 4m:
By means of this theorem we can separate from the rest
any number of terms, taken either at the beginning or end of a
given series.
1 = x”
10. Theorem. SS, 2") = ——.
1 a &
For, ve} = a} ( b=
1-wv@
anne a™
o . or nm
— S., ge} — S.. ‘pap a Sn = 5 (4) and (5)
y a—1 a” n—1 a” a
fe +8, — 5 i 9)5
1—w@ Pd ep it te fe ie (9)
1 = x”
- 1 —
a” —a*- 3
1. Con. 2S a" a.
a-—-@
1 n a”
12 "OnE: 2: = Sa") 4
1 Sod ees
eee a
anc = S,, =] m—1 s va} gs —1)* ; .
1 + v : ( ) ( l +: =
If w is <1, the term will diminish as 2 increases,
1 Sh
nm
and therefore, by taking » sufficiently great, §,,a"~' may
1
be made to differ from by a quantity less than any
1 = v&
assignable quantity, although that difference will never vanish
]
for any finite value of 7. In this case ; is said to equal
, —w@
* term is v"—!; and this
=
4 m— I
= Sn av .
an infinite series of which the m"
relation is denoted by the equation ;
Z — &
13. If the law which determines the value of a,, in the
n
series S,,@, is such, that a,,= 0 for every value of m greater
n
eo
than 2, we may substitute §,,a,, for S,,@,,; and this substitution
will frequently facilitate our investigations.
Qn n n
14. Theorem. S,, Qn = Sn @en_1 £ Sm Gems
2Qn—1 n n—l CF
a 1 x _——
and Sh, Qn = oa Gam — 1 ar Sin Ooms ‘ i,
iy ?
Py | ¢ A f
fk or, hn On = A + Ay 27 a3 + ay + O5ck ae + Dry, |
= (Ay + Ay + As + n_ 1) + (dg +O, +5... + Ao)
n n
| 1
= Se Gon 1 ar Sin Aon ©
2n—1
and Si Gy, = ay a ay ot a: ar vee a Asn)
= (A, + Gg + 02. + Uon_1) + (Ge + Gy + «2. + Aen_o)
n—1
n
\ 1
= Sin Qom—\ + on Gon:
C1 a a
1
Cor. ho a, = oye Azm—1 - oA om «
By means of these theorems we can separate the odd and
even terms of a given series.
15. The symbol §,,S,,d,,, denotes the sum of 7 terms of
&
which the m™ is §,@,,,: that is,
Tr s
8 8 8
Si Sn Fina = Sn Gi,n + Sn Gen + Spds,n be - + tS n
=O, + Got 3t+.---+&,;
+ fly, + Mo,9 + Aog+..-. + Gas
+ lg + 3,9 + zg +....+ Gy,
+ &. + &e.
+ G,, + G,2+4,3+.-..+G,5.
It is obvious that the same principle may be extended to any
number of symbols of summation.
6
s
16. Theorem. (Snadm) x (S,5,) = SnGn-Srbn*.
nun
’
1
For, Sp Gm = 0, +d +0,+... $4,.
.e (Sn Gm) x (S,,b,) = 1. Sy 0y + dy Sy by + Og. Sy Dq+ eee + Ay SnD,
r ‘
= of An S. 2, .
17. Theorem. If is independent of m, and s of m,
r 8 hl Rae
then Si, S,, Qn,n = S., Sn Gn, n°
,
8
For S, Gn, a Gn,1 + Bn,2 ats On,3 Ties at Bn, s ~
r r r r r
ve Sn Sn On, a Sn An 1 ap Sin an, 2 ap Sn Bn, 3 Ses SIP Sn An, 53 (4) and (5) >
’
fp
S,S8
a n m Lin, n*
oo © ry
Qa I ei) al
18. Theorem. Sn Sn Gn, n= a S., On—n+1,n°
o oo
=)
For, Sn S., an, n= S Gan n> (17) 5
ron) —-) ro) co
aI a a NI
=D An, tS, an, ot Sin Gm, 3+ eee +S, Am, at &e.
= Ay, 1+ Ae, 1) FAs; 7A, Fees FOn,) +&c.
FO) ots ots oF 022 tOn_1,2 +&c.
+4) 3+423+--.+Am 23 4-&c.
+4), 4+ -0et+On 34 +&c.
+&c.+ &e. +&c.
+4), n+KC.4+4n niin +&e.
+&e.+. &e. + &c.
+4), m +e.
+e.
* That is, the product of eg! multiplied by 5. bn, 18 a series consisting of +
terms of which the mth is am» Sn On-
7
2 m
1 3 oF
a} a! SN) Ql
=e, ntSn43—nat es, nt A5—n,2+ &e.+5, On—n+i1,a> &e.
by summing vertically ;
eo m
= Sy Same
o m 2 oe
i Cor, t. Siete HU to tt Mae
oe m— 0 om
1
20. Cor. 2. Sn Sn On, n= n,n SERS AU ae (9);
m
2]
)
=S
—Nm nOm+iin
mo 2
=n Da Gace (19);
co ok a LO °
:
€ oN aM
20 Phcorein. | S38, 0.0 Onan ee ae Oe lay eed ae
vi r
= ‘ - :
For, S29.@n,2=9nGi,nt9n2,n+9n%, nt «22 +9n Ont KC.
= Ay, +, oF Gy, g++. +A nt --- +4),
+ Ap, 1 +2, 9+ Mo, g+ o-- + Mo, nt oo. +A, ,
+3, 1 +s, 9 +s, 3+ 2. +s, n too. +s,
+ &ce. + &e. + &e,
NEC SGP atu Aare Saha Shea A
+ &e. +. &e. + &c.
therefore, summing diagonally,
= r
Sn S.@in,n= M1, 1+ (Go, + %,2) + (As, 1+ 2,24 hi,3) + vee
+ (Gy, +@y_1,2+@y-2,3+ vee +, ,) + (Gy 41,14 Gy, 2+ Gy —1,3+ vee +s, y)
+ (@y4014Ory12+4,3+---+4,,)+&e. + &e.
1 2 3 r
a 5S ,42—nnt+SnQs— nat Se Bes eee +8,4r—nernt
r r
“ts 4S ,Gr—n+ant ale ayaa &c.}
m
Y oo) |
ay NM
= Sn S,.Gn —n+l1, rt Sn S, Ont r—n+1,n°
8
mn m m—n+1
Al
22. Theorem. §,8,4,,=S_ S:Gn¢1=11
™m m—n+l1
and =s), Sree s,
mn 1 2 3 m 7
For, 8,8 ,.4n,:=S-41, 1+ 9742, r +9; Gs, + 022 +9;Gn,
=@),1-
+ Az, 1+ G2,2
+ Gs, 1+ 43,2+ 43,3
+s: +, 24+, 3+ My, 4
+Gs5,1+45,2+ As, 3+5,4+4s,5
+ &. + &¢.
+ Gin, 1+ Gn,2+ Im, 3+ vee coe tm, ind
therefore, summing diagonally,
mn
S291 On, r= Ay, t+ Mo, 2+ As, 3+ 06. + Om, m
+2) +3 oFs,3+ 0o¢t+Un,m-1
+31 +s 245, 3+ 26. +m, m-2
+ &e. + &c.
+n -1,1 + 4n,2
+Gn,1
m—1 m—2 m—n+1
ae a, PS, Oy 4i,rtOr +2, rt eee tS; Osan pcos +m,
mm m—n+l
So) SLU
and, summing vertically,
mn m—1 m—2 mnr+1
S,9;%,1 me a,, its; Gy +4, ot S, Ag 4.9,3+ coe +S, Qn+-r—1,n7 coe tn, m
m m—nr+l
=) Sie ee
9
2m m n
a | A
93. Theorem ioe eG. n + (eqs Hlog es ;, ,)
m—-l m—n
1S, Qom—7+1, reitOn S; (ag — 131, nt 1 Come eta 2) >
2m—-1 n m-l1 7
and S., S, Qn, 2 = S, (Git in r+ Gon —++1, rt) Qom —r—1, r+1
m—-—2 m—n—1 2m—1
z
+8, S, (@om—r—1, dna t-+ Ginn 1, 28a abet Dr Som 1 r-
2m
2m n 1 2 3
For, Ss, S,@y,y= SQ, +9; le, +S As, yt Pee BS. don:
=)
+ Qo, 1+ 2,2
+ G3, + 43, 2+ A,3
+4, 1+ Us, 9+ My, 3+ %,4
+ Xe. + &c.
+ Bom, 1+ Bam,e+ ete + Bm, 2m 3
therefore, summing diagonally,
2m n
S,S1@n, r= ( (G11) + (4,1) + (43, 1+ Ae, 2)
+ (G4, 1+ 5,2) + (Gs, 1+. @4, 2+ As, 3) + (5,144,243)
+ (Gy,14+ 5,945, 3+ Ay 4) + (Ag, 1+ 47,24+%,3+%s, 1)
+ W&e. + &c. +&c.
+ Bam —1,1 + Gem —2,2+ Bam —3,3+ Gam —4,4 + Vom —5,5 4 v2 2 + 4m, m
+ 2m, 1+ Com —1,2+ ham —2,3+ Lom —3,4t bom —4,5+ +++ F4m+1,m
+ Gam, 2+ Lom —1,3+ om —2,4+ Gem —3,5+ Com —4,6 + e2¢ + Omti,mt1
+ Gem, 4+ Gem, 1,5 + Gem —2, 6+ Gem —3,7+ Dom —1,8T «+» + Um+2,m+2
+ om, 5+ om —1,6+ Lam—2, 1+ Vom —3,3+ Gem—s, 9+ 00+ + Om4s3,m+2
+ Gam, 6+ Com —1, 7+ Cem —2,8 + Cam —3, 9 + Cam —4, 10+ OL +n 43,m4+3
+&e.
+ (@2m,2m—3+ em — 1,2m—2) “Fe (Gom,2m 2a Bem —1,2m— i)
: + Gam, 3+ em —1,4+ Lom —2,5+ Lom —3,6+ Gam —4,7+ ++ + Um+2,m4+1
; (Giana) ar (Gin, 2m)*
B
10
mn m
S15) S, (Gon —y, 7+ Oon— 14 ir) +8, Qom—r+1, 441
m—-1l m—n
+S, Ss, (Gm: venti t Com—rtirente) 5
2Qm-1 n 2nm—-2 n 2m—-1
and S, S, @n7=9n S,GartSr Qom—1, 79 (9) s
m—-l n m—l
=S, S, (Gop y,r+ Con 14 “) +S, Gom—y—-1,1r+1
m—2 m—n—1 2Qm—1
+S, S, (iis aes Bete Meee cpa a ahaa) Pm pOann ae
by the former case.
n—1
24. Theorem. Vf Gy 44=Gn+0_. (Ca, then a,=a,+S,,0n-
m=n—1/)?
n—1 n—1l n—l
For, Sn Ne Sn On+ ros De. (4) & (5).
n—2 n—-2 n—l
: y S
m Gm +1 au: a, = ay ati S.,. An +1 a 5 b, > (9) 2
n—1
d,=4+8,,6,,, cancelling identical terms.
25. Theorem. If a,4;=Ccan+bn, (as
m=n—1/}?
n—l
— =u m—1
then =" . a, +3, Cc » Des bs
For, 1 APR 3 tt) 3
OP Gazi =C" Ogg 0" 7 -- Dp, multiplying by c™>);
n—l n—l n—1
and Ss ORG en COs Seth hm a Deen (4) & (5);
n—2 n—2 n—l
Y = =
y+ Sn ce Gy, —-m= =P c™ a, —m 1 c : : ay AF oF ce" b, —m, (9).
n—l
5. Wi pe ney + no CED.
n—-m*
26. Theorem.
se = at
x ome
(Sn Gn-1 bet ") (S,, Ba, a ae ') ae S.. a
11
m
= oe
For, (S., An-1 anit} (S,, baw Bi ‘)
2g. Con. iS Ona -
8
co
a on Bn —-1- ee S,, by ap
oe ©
7 Sn Sh Gy 1° ee . ak a
i=)
= Sn S,, Qn—n 5 Oe 6 are
oo m
-1
Ee eT ae S,, A BY s:
2 m
ree i ot S., ge. S, Amn —n+ G—1>
1 \
7 S, an—n °
Goats
(16) ;
(6);
(18) ;
(6).
CHAPTER II.
ON PRODUCTS AND FACTORIALS.
98. ‘Tue Product of n factors, of which the m" is a,,, will
be denoted by P,,,a,,; that is,
nn
Pose, =a). 85 dg...2 @,,.
The symbol 1D a,, will denote that the 7*" factor is to be
omitted.
29. Theorem. If dm =bn, (=), then Pram = Prbn-
m=n
For, since a, =},
a, = by
a, = b,
&e. = &e.
a — bs.
therefore, multiplying,
Uh © a 21g ne (Og = Ui 05. One. Ons
n
or Ps an = Ps Dn-
30. Theorem.
If b is independent of m, then P,,(a,,b) = b". P, Dns
nt
For, P,,(a,6) = a,6.a2b.a3b...a,6.
UjQs0nee.
n
n
= b” > Pp Qn:
13
31. Problem. To invert the order of the factors of a given
product.
7
Now, Paqe— a1 45.0;
o
coe A
n
= d,,.,_,-4,_2...4,, by inverting the order of the factors;
n
Pe Qn—m+) 0
If, therefore, we substitute »—m-+1 for m, in the expression
for a,,, the order of the factors will be inverted. See Art. 8.
n—Tr
32. Theorem. ie an = Ba (Gn) 2 P,, (@,4:m)-
n
For, Ps an = (a - Az + Agee. a,) (G41 »Qry2+ Ur43-- Gy)
m—r
= P, (CA) 6 P,, (G5).
By means of this theorem we can separate from the rest
any number of factors, taken either at the beginning or the
end of a given product. See Art. 9.
n+r
oo SCOR. jee (4) : ] ye (@, 5m) =pe An:
0
34. Theorem. P,,a, = 1.
O+n
0 n
For, ipa (an) *) Ps (a5; =) = | Ons (33) 3
0 n n
or ie (4) C ) Pak ay, = PS Ans
0
. Pia = 1 by division.
—n 1 1
35. Theorem. P,, a,,=—~——— =
n we yc -
Be Am —n i a_ (m—1)
n—n —n
n
For, mOn—n = | (Gn_n) c | a Ons (32) >
and = 1, (34).
—n n
a jie (4,,) $ Poe = 1 3
14
—n
and P,,a,, = ,» by division ;
n
m&m—n
ee
m @_(m—1)
n—l
36. Theorem. - If dy.) = m-b,, then a, = a,.P,,dn
For, 6,, = ;
n—] ait Qn +1
oe |i = Ps ( ) ? (29) ;
m
n—2
pe P. (Gn +1) .: a, ; (32);
a - Pre
a pie, ae
= —, cancelling the identical factors.
n—l
a, = ay, . Re be
37. Theorem. If an,.= dans.
n—l n—l
then Ayn = Ap. P; bem and Ayn) = A - BP; Bom 1:
Bom+2
Kor —
Gam
n—] n—l Qo me
-s P Don, = ‘PF ( ) b) (29) ;
‘2m
n—2
— Palaons2) don (32) ;
My : ‘Ps Aon 42
a, :;
= —, cancelling the identical factors.
Ay
n—)
Cyn = Ap : ip Bam:
Gom+1
Also, Gyan ae
hy m—1
lI
n—1 n—1 Doms
es Ec. Bom —1 = Pa ’ (29) ;
‘2m —1
n—2
ps (aon4) © Aan} : (32) ;
Q ‘) P,, Bony, +]
Ag, 1
a,
n—l
"o Aan) = ay * P, (Oam—1)-
38. The symbol |a denotes the product of m factors
forming an arithmetical progression, of which the first term
is a, and the common difference m; if m=—1, the m may be
omitted; and, if, in the same case, m=a, the n also may be
omitted: thus
|@ = a(a + m)(a + 2m)...(a +m —1.m),
n,m
|a =a(a —1)(@—- 2)...(@—m + 1), and
n
|a=a(a—1)(a-2)...2.1,
39. Theorem. ab=b".\a p
n,m m
For, |ab = ab. (ab + m)(ab +2m)...(ab +n —1.m)
n,m
ll
S
g
LS
g
ah
\s
ae
as
g
aie
SS
3
eet
aa
g
oe
=
|
x
3
a
40.
For,
16
Theorem. \@ =r ig, Sale an1.
ym ny—m
|@ =a.(4+m)(a+2m)...(a+n—1.m),
n,m
= (4+n—-1.m) (a+n—2.m)(a+n—3.m)...(a+m) a,
by inverting the order of the factors;
4].
For,
For,
43.
For,
=|a@+n—1.m. See Art. 31.
Nn, —m.
Theorem. a = |a .|a+trm.
My, ™m Vv, mm u—?r, ™
|@ = Ja.(a+m)(a+2m)...(a+r =) -m)t x
n,m
\(@t+rm)(a+7r+1.m)...(a+n—1.m)t
= |a. |@ +rm. See Art. 32.
r,m N—r,m
Theorem. |a=
0, m
[a \a +0.m = \@ 5 CAL ys
0,m n,m O+n, m
or \@ . | = |a.
0,m nym n,m
4 ja = 1, by division. See Art. 34.
0,m
Theorem. |@ = eS = SS.
\@ —nm a—m
‘ —Ns™m™
n,m Nn, —m
a—-nm = |@ —nm.|a, (41),
m—n,m n,m —n, mM
and =i, (42);
-|a-nm.|a=1.
n,m —n,m
, by division ;
, (40). See Art. 35.
44. Theorem. —— =0.
1 1
[== [==
—m,—1
= |—m+m, (43) ;
m, —1
ie
m
For,
=m);
45. 'The theorems in Articles 34, 35, 42, and 43 are ana-
, 1 ‘ :
logous to the equations a°=1, and a~”= —; which last equations
q”
indeed may be deduced from them as particular cases.
n—m
EN
™m n—™m
48. Problem. 'To shew that i is a whole number; 7
m
and m being integers.
m+1 = m+1 aa S,—, (24)
m+i1 m+ 1 m
T
n—1
=S,—
r | m
n n
If, therefore, —- were an integer, —“t would be an integer ;
[= bes
n n
but fr is an integer, and therefore — is an integer.
m
te
19
49. The symbol § ay +b, §...§¢}...¢ denotes the re-
m m+1l n+l n+l 1
sult of the combination of the symbols
ay ty aa + be fon fy + By} Of oof 5
1 2 3 ntl ntl 1
the brackets being omitted after the expansion, if they are then
without signification.
m—1
50. Theorem. }ay +055... Cesiee a, Pb, +e: P,6,.
m+l1 n+l
For, t AmtDm § ae. {C= {a, +b, fa.+b,§...$ 6 $...$
m+l n+l 1 2 3 n+l ntl 1
=fa+ D+ 3b, eas
1
+ §[B,y00.by-1- Ay §B,b2...b,¢} ...}
n+1 n+1 1
= a,+b, a+b, bo As .e.
+b, b,...6,_,@,+6,65...6,€
m—1
Euan . P,b,+e. PG.
51. Theorem. If a,=6, + €,- +23
m s—l
then a,= S, a Piensa. at Anime: Prokes. ar
For, a,= SOO, Dn at Cn za) see nen a see Carnes a
1 2 3 s s+1 m+l m+l 1
by substitution ;
™m
= bn Fay Os ante NU eee (49) ;
s s+] m+l
m s—1 m
= Cera meee et Onemaces Cn+ial a (50).
52., ‘Corsi lfias = '0in 4 Cnyn “Gate dees; then
s—l
Am, n= S, i eter a,n+s—1.8 * Pi (Cnt. a, n+t—l. a)
+ Ans rantrB ° P, (Cer oat 8)-
CHAPTER III.
ON COMBINATIONS AND ARRANGEMENTS.
53. Tue symbol Ce: will be used to denote the sum
of every possible combination, (without repetitions of any one
letter in the same combination,) that can be formed by taking
m at a time of m quantities of which the r™ is a,; and the
mn; s
sym bol C,a, will denote the same thing, with the condition
that a, is to be every where omitted.
m+1,n+1 m+l1,n Mm, 7
54. Theorem. C,a,=C,a,+a,,,.C,a,.
m+1,n+1
For, C,a, must consist of terms into which a,,, does not
enter as a factor, and of others into which it enters as a factor
m+1,n
once only; and it is obvious that C,a@, will express the first
mM, nN
set, and that a,,,.C,a, will express the second. Hence the
truth of the theorem is manifest.
55. Theorem. If b is independent of r, then
my, rr Ms, 7
CHG Oa 0" Cras
m,n -
For, C,(a,6) denotes the sum of a certain series, each term
of which is the product of m quantities, and into each of which
My, rt
quantities 6 enters as a multiplier; and C,a, denotes the sum
of a series, each term of which is the product of the same m
quantities, each being deprived of its multiplier 6.
56. Theorem. If ais independent of *,
ye
mn
then C, (a) = . a”.
m
n
n,m
For, the number of terms in C,a, is Ee and each term
m
myn
consists of m factors. Since, therefore, in C,(a) each of these
factors is equal to a, the truth of the theorem is manifest.
57. Theorem. =(Cigse
n
Pia
For, the numerator of the first member of this equation con-
sists of every possible combination of m quantities, taken m at a
time; and, hence, that side of the equation consists of a series of
fractions, the numerator of each being unity, and in which the
denominators are formed by taking away, in every possible
manner, m of m given quantities, and will, therefore, consist
of every possible combination of these m quantities taken »—m
at a time.
0,2
58. Theorem. C,a,=1.
0, N—nN,n
For, C,.a, = C,a,
m,n—m
59. The symbol C,., (a,.0,) denotes that there are 1
quantities of which the 7 is a,, and m others of which the
s™ is b,, and that every possible combination, (without repe-
titions of the same quantity in any one combination,) is to
be formed of the first series, by taking them m at a time;
22
and that each combination thus formed is to be multiplied
by ~—m quantities of the second series, so taken that in
each of the combinations the whole of the natural numbers
from 1 to m shall appear as indices: thus,
2,3
C,,5 (4.5) = @, d2b3b,b; +; A3b2b,b; + a, d,b2b3b; + 4, A5b2b5 0,
+ a,03b,b,b; + d.a,b,b,b, + a.a;b, bb,
+ a; a,b, 6.6; + A350, b.b, + a,a;b,b,bs.
60. Cor. If b,=6, then C,,(a,.0,) =0'-".C,a,
61. Theorem.
n—m+1,m n—m,m n—m+1,m—1
Ci (4, : bs) = An41- Cs (a, , b.) mL Dna q Cr (a, ; b.).
n—m+1,m
For, C,,, (@,-6,) will consist of terms into which a,,,, enters
as a factor, and 6,,, does not; and of others into which 6,,,
enters and a,,, does not. Also each of these terms must con-
sist of m factors, exclusive of the factors a,,, or 6,,,; and each
of them must contain 2—m-+1 factors of the series @,, d2,.¢+@,+15
n—m+1,n
and m of the series 6, 6.,...6,,;. Also C,,,(a,.6,) must con-
tain every possible term that can be formed consistently with
n—m,m
these conditions. Hence a,,,.C,,,(@,.6,) will contain all the
n—m+1,m—1
terms of the first kind, and 6,,,.C,,,(@,-.6,) all those of the
second kind.
62. The symbol A,,,a,, denotes the sum of every pos-
sible Arrangement that can be formed of any number of
quantities of which the m™ is a,, these arrangements being
subject to the condition that the sum of the indices subscript
shall in every single arrangement amount to 7; repetitions of
the same letter being allowed in any arrangement: thus
4
AL Ay, = Ay + AzQ, + AgAy + A,H, QA, + A, A, 4, a, + A, A, Ap + A,
= 1, + 20,0, + A.” + 2a,. 0,7 + a).
23
n—m
63. Theorem. A,,@, = SnGmn-
+r Oy
n m=—1
= S., Qn—m+1 . ‘AG Dy.
For, A,,@, is the sum of all the terms that can be formed
of any number of quantities a), a, &c. such that the sum of
n—m
the indices subscript shall be m; now a,.A,,a, will include
n—m
every term in which a,, is a factor, and §,,a,.A,,a, will
include all the admissible values of a,, and therefore every
n
term of A,,4,.
= Se keer ars (8).
64. Cor. AG; =r
n n
65. Theorem. If a,=S,,0,-m-6,, then a,=a,.A,,0,.
1
For.) 4) = dy. 0, = dy) iA 4, 5.
Qe => a,b, + Ay b.
2
= a, (6,? + 6.) = a. A,,0,-
as = a,b, + a,b, + a,b,
= Ay (b3 + b, bs + bb, + b;)
3
= )- se 63
a, = ab, + Az b, a a,b, oe Ay by
= dy (b,* + b,? by + bbb, + bb; + b2by” + b” + b3b, + 6,)
4
= Ao ° AC b, .
n
Suppose, therefore, a,=a,.A,,6,;
n+l
then O41 = Sa Gn—m+) On
n+l n—m+1
Se nbn An 0,
n+l n—m-+1
Ce On LB On, (6) ;-
n+l
Gj Aaa, Ors (63).
i}
24
If, therefore, the law were true for 7 and all inferior integers,
it would be true for 2+1; but it 7s true for 1, 2, 3 and 4, and
therefore for n.
n
66. Theorem. If a4,=¢,+SnQnr—-m-Oms
n m—1 n
thena2="S,, Cro 54)- AG; Or + Gp Ag Oe
For, proceeding as in the last Article, we shall find that
m—1
=s. C5_m* Avy; +. A.,b,:
m—1
n
Suppose, therefore, a, = Sy C,—m+1-A+,5, + %- Ay ee
n+l
then Qn41=Cas 1+ Om e Qy—m+1
n+l n—m+1
=Cgh ita On sor eheea 1 een Was at oar 728 by substitution ;
n+l m—1 s—l n+l n—m+1
=CriitS, hs pies Cn—s- A, 6; + - Ss Dm ce etO;5 (8) and (6) ;
s—l n m s—l
=Caiitbn41- S, Cy_s- A,,b, +S Ds tea Sila sua eur Or
n+l
+a,.A,,6,, (9) and (63);
s—l n+l
=e ae m+1° S. Cm—i+1 = Ay Opty? A,B,
nn—m+1 s—l n+l
=A +S Ss0sin serena 0,-+a5- AG 0, (6) and (22):
n n—m+1 n+l
=CatitSm Om oF Oye —m—S+2° (A. Bi4iay. A 4,5, (6) ;
n n—m+1 n+l
=n Cm - Ase b, +, . Aj b,s (63) 5
n+l
Deeg Se Ch—m+1° A Bet dy. A i Ors (8) ;
n+l m n+l
Ss) Ch—m+2 ° Ase b, +a, 3 A... b,, (9).
If, therefore, the law were true for and all inferior integers,
it would be true for +1; but it is true for 1, 2, 3 and 4, and
therefore for 7.
CHAPTER IV.
ON BINOMIALS AND EXPONENTIALS.
a Te m—1
67. Theorem. P (a, +,) = C,,; (a, - 8,).
For, by actual multiplication,
3
P, (a, + 6,) = @, yz + A, dgby + A, O30, + Ay Ayb,
+ A, b.b; + debi bz + a,b, b, + b,b.b;
4 4—m,m—1
= Sn C,,s (a, : b,).
Ss n—m+l, m—1
Suppose, therefore, P P. (a; + b,) = C,, :(@;-.5s)s
n+l ict n—m+1, m—1
then P, (a, + b,) = (Qn41 + bi4:)-S C,,5 (4, - bs)
a+l n—m+1, m—1 n+l n—m+1,m—1
= Sin Gn41+C,,5(4,-b,) + Sp O41 - C,, (a, UP) (6);
a
= Anir- ¢ GBS b,) a S. An+1- Ca, 0.)
n—m+1, m—l
ay Dun. Cn (az. bs) + Ones C 3 (4,. b.)s (9);
n+l, 0 n—mMm, m n—m+1, m—1 0, n+
=C;, (a, b)+S, 5 Gait rigs (a, 6) +8, 41 C,,.(a,- bt+C,, (Gy Bi} (0): ;
n+1,0 n n—m+l,m
= C,,.(@, - b)+5n C;; (a, 53 bs) +C,, (4,.. bs), (61) >
n+2 n—m-+2, m—1
=S, C,(@,-5), (9).
If therefore, the law were true for m factors, it would be
true for n+1; but it 7s true for 3, and therefore, it is true for 7,
D
26
S n—m+1, m—1
68. Cor.1. Put 6,=a, then P, (a@+a,)=S, C,,.(a,-2)
n+l n—m+1,n
= §,,a"-'. C_a,, | (60)-
69. Theorem. If a, is the r™ root of the equation
n+l n—m+l1,n
0=§,,@n_,.a"—}, then shall a,_,=C,(—a,).
Foret. ae EPG)
n+l n—m-+1, %
= Sn: Ege . C,(- a,)s (68) 3
n—m+l1, n
an-1 = C,(- a,)-
70. Problem. Given b,_,;=S,,4,.-'. am, ("Z,), to find a.
n—7r,n;t
Multiply both sides of the equation by C,(-—a,)
n—r,n;t N—7yn;t
then b,_,.C,(-4,) = S,, Be Glo. CO =a,), 6):
m™
n—r,n;t
7b,_,.C,(—4,) = Sy.2n-S, men, (-a), (4) and (17);
as
n nst
= Sn Xn - E. (4, a a), (68).
n;t
But P,(a,,-—a,)=0, for every value from m=1 to m=n,
except for m = ¢;
n n—r,n;t nt
therefore, S,b,_,.C,(—a,) = %. P,.(a,—,),
M—, rst
S, b,? ee C, (- i)
mst
P, (a, ry a,)
and a=
27
x -) Wit
S.67-!. —a
vale Cor. If Des; = bi, then a n;t oi 2,
P,(a,-4,)
nit (b—a,
op ( _ Pa (G3),
a, — a,
72. Theorem.
n+1 n+l
S a yma} n n—m+) S a b"-)
m = Ihe Es =i {S ‘m—\*
m = S,,2” eS Ge eh" eT a 1
ee b ce b
a} 3 6" - m—l
For, “> =S,a".b"-""',, (11);
a Sen a ae oe m—]
s. m ; aes ; =O, \- Seo Ee :
ae Ci
n+1 n+l
it m—-1l n+] m—1
ees are SnGn—15 1 = -r-1
and A ys hada om fi) hee Folie = oe ples Bt if hist 9 (4) ;
0 % m
a a = al = —r
= 4).5,0°°'.b-'+S,,a,-S,07-'.b"-", (9);
n m
= Sm e S,a77! é bn-7
n n—m+1
& x =
= 8,20". S,dner1-6', (6) and (22).
n+l n+l
—] n a—m+1 m —1
aU ae bile a ay Sinai oU
ay =S,07 la tr 08 1 Se PDS
n+1
73. Cor. If bis a root of the equation 0=S,,a,_,.2"~',
the second side of the equation is divisible by x6.
For, the remainder after the performance of this division
n+l
is Sp@m—1-6""!; which =0, since 4 is a root of the equation.
28
nm
Ma
3, =x .|@ : |b: ; n being any
n—m+1,*+ m—l1,7r
74. Theorem. \a+b
positive integer.
For, Jerbma+d= ao bid
sr 1,r
a+b=(a+b+r) (ja+ |)
ar i597 > el5
we
=e Gir n Pas)
lr
Lats (Be3- a.\b+ 1)
1, lr 2,7
=[eeaee DP a. Garb) |o(@siaH
297
eee
=|a+|a. [b+2.|a. b+2.|a. |b+|a. [+].
37 2r Lr 2,7 lr lr 27 1 %r 37
=|a+3.|a.|b+3.|a.|b+|b.
37 2r ir lr 27 37
sae
=Sa a. [a |b
tL __
|m—1
Similarly, eS S a [dete Leds
il 5-—m,r m—1,7
n
Suppose, therefore, ee at
ae
a n—m+1,r m—l,7r
then }a+b=(a+b+mr) .|a+b
_
= Sp Za! je, |b -(a+n—m4+1.r+b+m—-1.7r), (6);
n+1
29
=Sr—=.(ja .|6 +a .|b)
|m—1 n—m+2,r m—l,r n—m+l,r mr
n nm Le
= Cae [bse |e |b. ee oe 2 [6
nt+l,r 0,7 | m n—m+l,r mr c= a n—m+l1,7 m7
paige k (9) ;
n+1,7r
Lavina
=|a_. oS. (E+ ) |e \b+1a. |b
n+l,r. 0,7 = n—m+l1,r mr 0,7 n+l1,r
n+1
=|a .[b+Sp2—.|a .|b+|a.|6 , (47)
n+l,r O,7r | m n—m+l,r mr 0,r n+l,r
n+2 | +1
Beles. fas 1b (9).
Tan | [o
m—1l nomt2,r m—lr
If, therefore, the law were true for , it would be true for
: but it is true for 4, and therefore, for 7.
1
7 \Cor: 1. ne .|-b.
1 n—m+1,r m—l,r
But bali (=1)" Ae » (39).
m—l,
n
i
©) Lee S.C)" ere
1 n—m+l,r m—1]
: Ole
46. Cor. 2. - Since |r=|m .|[m—m+1, + eee aaa”
ee m— |
ceo A a ee
and n,r —= Sn (#1) . n—m+1,7__m—l, +r
ic ia
77. Theorem.
> | eae eae z [o . a"? = |a+b. a"
m—1,r n—1,7r ew m—1, rT
(3. =) (8 |.
eer [ee
m—l,7r n—1,7r
For, (S ye [z-1 )
a |b
io=) ——— 5
=S,,@ m— 1 5 ee a, (26) ;
|m—n. |~-1
_ lett
=S,a"-) mht, (76).
z)
|m—1
m—-l n m—1
F |@ - &@ a |ma - &
78. Cor. 1. (Sa =), ;
|m—1 m—1
n being any positive integer.
: a—b.a”-! OP ent coats fy agy ae
moms (REBT) (Abe) ale"
= =I =
‘ |a ea 4 |b a t |a—b. m1
S,, m—1,r S, n—1,r 2 Sn m—1,r
m—1 nm—1 m—I
31
Sie Cant (Ss. [a pel |= | (Sele. ae :
t oon
noe nfo s gf te J [m—1"
82. Cor. 5. Put S. a “ai SO
Oe ee
then $ f(a)? -" aaTOye =) =f(0-na), (79);
PatGee)
83. Cor.6. {f(a)t-'=f(-a);
t ag”) t =
84. Theorem. (Sx a
m—1,r m—I
+terms in a.
=
ih
iF
z
t gm} t gy)
For, (Sela. 55) ($2)
| a 6
m+n—2
@
(16) and (6);
aa
t ———
= S,, Spee te) + termsane, “(18);
ees
BSS
t
Sp eee terms in a, (6);
‘nan [n=i
, lord. am’,
= Dee terms an 7, 9 (76):
|m—1
, a a gm} n , na a” 1
85. Cor. 1. (s, == _) = S, 22 a termsan ae
m—1 m—1
a+1 |
86. Theorem. (a+b)'=S,,——.a"-"*).b"-'; nm being
[mar
any positive integer.
n n+1 m—1,n
For, P, (a+6,)=8,,a"-"*!.C,(6,), (68) and (8).
m—1,n
Put 6,=6 then (a+b)"= tes Otte O2(b)
aa
n+l
=S, mb a4} b"-1, (56).
"|m=1
This theorem may also be proved as follows:
n+1 oe
a+b=S,, |e |b 3 (74).
nT "Tm é 5 n—m+1,r m—1,r
Put r=0, then |a+b=(a+6)", |@
=o" P41, and |b =pr-1,
sr n—m+1,7r m-1,7r
n+1 n
*! (a+b)"= =S, a7 m+1 _pr- 1
"[m=1
33
87. Cor. 1. If we invert the series we shall get
n
+1
(a+b)"=S,, en an. Baa (8) ;
| 7 |
pie =p m—1 ; (46) :
M Jaamei mel m—1
therefore the coefficients are the same when taken in an inverted
order.
88. Cor. 2. Since |2= |7 - |2—m-+1,
m—1
Co ry qr-m+l Bm}
[nT [acme |e
89. Theorem.
0 In
(a+b)'=Sy ZL. (aby"—" (a4 4 Bt es (aby, or
[mi git
4 (n+1)
=S,, 3(ab)"7 1 (as aa 7 seth Ve
‘bet
according as ” is even or odd.
n+1 |
For, (a+b)"=S,, =~ .a"-"*1.b"-!, (86).
|m—1
“(1)_ Let 2 be even; then
n VL)
Bie ia n—m+1 m—1 tn tn
(a+b)"=S,, a + =<. (ab)
m—1 oy
1%
+§,, — 1 Oe i (9), (8) and (46).
34
= S22 = . (ab)”~ ; (a"- Paez + Us mt ae E ll
[m= 2”
(2) Let be odd, then
$(n+1) | $(n+1) |e
m— : a™-} f pat. (9),
(a+b)"=S,, [ea ett Ont +n Sate
= m—1 = (abyr ary —2m +2 we fie 242), (5)
90. Cor. If m is even,
n
(a- b)" “5 (2 1h) gas = (ab)”"- (ar. 2m+24 pn- 2m+2)
xy ap) 28 by".
+(=1) ED (ab)™;
and if is odd,
n
$(n+1)
(a- —b)” =S,, (]1)?> a eed = (ab)"- 1 (a —2m+2_pn— wei)
ee De Tia
91. Theorem. ( + a) r= S., E a ‘ [ : 2 n and 9°
m—1
being any positive integers, and @ being less than unity.
© rm—-1l\ +r & Puen
For, (S, +t.) S,[m .——, (78) and. (82) ;
m—l
n+) [a
=S,, — : Y ime (13) 5
|m—1
=(1+a)", (86).
m-'_.a™-1, where m is any
93... Cor" papost
|m-1
rational quantity ; w being less than unity: (86), (13) and (91).
2S Nay | Be
93.;_ Cor. 2. " 1 + #)F=5, F
rr) |m—t1
m—1
=(S tyr).
m—1,n"
ae
m—1,n
wt (140) =148,(-1 Le Ta (=).
1 o
95) Cox.4.. (14+2) 28, i
=S, (-1)"- a (7). (39).
96. Theorem.
3
(5. ae (5 —).8 oe a
o a7} am-l roa) b’-1 g-} © m an”. pr}
For, (S, ~ | (Ss, —) Bf ole i Sea aaa
[mar IW [aa [mn fani? O°
2. (a+b)"—! ym—1
nm m—1 y) (88)
97. Cont. P, fete ae Ona
|m—1
98. Cor.2. Put a,=a,
o aq™—1 gr} n © na m—-1 aml
then (8.7) =S Slag a 8
[ma [ma
o m—1 m—1 my kes m—1 m—1
99. Cor. 3. {8. (<) i ) 6.
mi ee
l
(8, a” 1 ym a -8, ( am: a} |
(ae ee
100. Cor. 4.
ond (a—b)*=). a=! ° GB -). a) 2 a™—1, ya!
(S, a (s. =~) +§,.— 3
|m—1 n—1 |m—1
co a™—) , am) o b-) , at} I (a—b)” ~ at}
oe ea
m1 [2-1 |m-1
LOLS .Cors a:
(S. (Cote =<) (5. rm =~) a (a—a)"™"* a"! ue
|m—1 |m=1 m—1
(s “_<)\" 8 (a)? a"
m m1 =Wm pa :
402s1'+ Core 6.
is. qq") . al" aI8, (Ee) et =
m—1 |m—1
=o pay" a: an}
=Sn mi 5B eCEOL) = yt
and
1
(s —— q (s ae
" |m=1 wikis |m—1
(27 ay ae
z q
=Sn m—1
22 (42 ay
© a" -)
103. Cor. 7. Put z=1, then (S. ) Gives Mat ca
m—1 m—1
104. Cor. 8. Put a=./+1, then
m =VOm
m—1
where » is any rational number.
m—1
is ae 5 Vv #1)"
38
. 2 1 . .
The series S,, [m—1 occurs very frequently in algebraical
investigations, and therefore we shall use the symbol e¢ to re-
mye Ae ar:
present it; while «v1 will be used to denote S,, ;
m—
Hence the above equation may be written
= (ef +1)"")
gaye
m—1
105. Cor. 9. Let # be irrational; and suppose y and z
are two rational numbers very nearly equal, such that «>y,
and #<2.
Then e¢*, e”, and e’ are in order of magnitude; that is,
m—1
o sg o yn
S , e”, and S,, , are in order of magnitude.
oy ea er
m—1 a™ -l o° y”™ -l
ee , are also in order of
But S,, ——,
|m—1 m—1 m—
magnitude however near the values of z, and y are taken to
—_
that of «;
2} gst
7S
wd ie |m—1
106. Cor. 10. Hence, whatever the value of 2 may be,
ou ti?
we shall have ¢’=S,, .
|m—1
al 3] : m—1
107. Theorem. a*= pe Oe a
For, a=el%'**;
“ By log, a is denoted the logarithm of a in the system whose base is ©.
39
of ata '8.-4
a (a log.)
eee)
™m |m—1 >
108. Theorem. (a+b)’=a°+2ab+b .
(106).
This will appear from actual multiplication.
n-—m
109. Theorem. (Sna,)?= =S_4,2+2 ice CES Panes
n—r+l1 n—r
For, Sin Grim —1=2r+Sm br m- (9) 5
n-T+1 n—?r n—r
tse. Dy +m ye =a, 742d, Sn ete (Sen Gees (108) ;
n n—r+l1 n— n n—r
and S, (S,, ar4+m— y= Mice a, £28, 4,.Sy0y én +S,(Sy. Oran),
(5), and (6);
n—1 n—r
. (Snap)? +S, C.2))7— =S rdy “42 S, a, (Sia,
n—l1 n—r
+S, (826-20) (Sia, >)
n—m
(Sh Gn)? =8,, 9? +2 eh Dn S, Ant
(4),
(9) 5
CHAPTER V.
ON FINITE DIFFERENCES.
110. Is p(w) is any function of w, then Pp(w) will
denote the same function of @(w). This last is expressed by
p(w); and, the same notation being extended, we get the
equations
pg” (u)=p"*'(u), and pf" (u)=p"*" (wu).
111. Con... $"(w=$""(u)=9"(w).
-. P'(u)=u.
112, Cor. 2: o-". hb" (w=p-"*"(u)
=" (u)
113. Definition. If p(w) is such a function of w that
pl(ut+r)=h(u)+G(v), then P(w) is called a distributive func-
tion of wu.
114. Definition. If d(u), and wW(w) are such functions
of u that py(u)=\ p(u), then the functions p(w) and W(w)
are said to be commutative with each other.
115. Instead of p(u)+ (wu), it is frequently convenient
to write (p+W)w; in which case the latter expression must be
carefully distinguished from the product (@+W) xu, and must
be considered merely as an abridgment of the full form
p(u)t+(u).
41
116. We shall express {(@+W)(p+W)}u, by (P+wW).u,
and {(P+V(P+W(P+w)}us by (P+y)su:
that is, (P+) w=(u)+V(w),
(Pr u=G}(PtWyy + (G+ Wu,
and, similarly, (p+y),u=P(P+Wn1v+WP+Wnat
n
117. The symbol }(@,+y,)}u, will be equivalent to the
r r+l
expression
S(gitW)(Go+ Wr)... (Prt) iu. (See Art. 49).
118. Theorem. If (wu), .(u), (uw), &e. and Y,(w),
W(t), W3(w), &c. are all distributive functions, and commuta-
tive with each other, then shall
be We) fu=Ba)" “Cupp iu
For, (fit \i)u=i(w+v(w), (115).
(pet) = (pat he) Shi(w)+W(w)i, (117);
=. 5 h,(w)+Wr(u)t +Wo$gi(u)+yr(w)?, (115) 5
=pog Wt oynW+po(w+ Wyn), (13);
=piplu)+orplu)+ gone, (114).
S(t Wo) $x (ot Yo) pled) + useles) +p)
+Wiy(u)}, (117);
=) Piplu)+ Pry.(u)+ paw + vil}
+ Wi Pip(w)+ ov) + orri(u)t+Wirr(w)t, (115);
E
42
=PiP:fs(U) + PGs (O+ Pepi + oye)
+ PiPys(w)+oows(w) + Gris(%) +Vivews(u), (113) & (114);
=(hihspst Pipost Dipset Pepsi t Pest Privs+ Pair
+Wiows)u, (115);
4—m, m—1
= Sn 5 C,,s(br-Ws) ' U.
Suppose, therefore, H(petyh) ueS} Calay ies
then Voy fun (dissident epi paN any
HG FS.4° C, “CG. pw] + Wiel Sah “Cie ay)}u], (115)5
LNT pis aro SY aErneenperine: eer W)tw], (113);
Si dO GN EPS IC. tote (210)
n—m+1, m=)
= Sa C ue, CSV OCG (bey) } (118);
“56 Cab V) Badale GU BN ert,
Hen Gulu Os
=16.,G.91)4+ Bal purr CalQe) Yon Gee]
PEG} ©) and (114);
jE G+8. Cages Galgeydiuy (1):
-{8. EG ©):
n+2 rte m—
=S,,} OC My)}u, (115).
43
If, therefore, the law were true for », it would also be true
for n+1; but it és true for 3, and hence it is true for 7.
119. Cor. 1. Let the functions p,(w), p.(w), &e. be all
similar to each other, and to @(w); also let W,(w), W.(w), &e.
be all similar to W/(w); then
(pet yr) ju becomes (+ ),#, and
eis Base | 2
n+l n—m+1, m—1 n+l
Sn i C,,.(p,..,)} w becomes §,, ca Mg oN aa
(56) and (60) ;
mean ae
Pe rr a
ror” Con. 2. Butt gp, and yy denoted quantities instead
u+1 [me
of functions, then would (@+W)"=S, ——. po "*!W""!; and
*[maa
hence we may express the preceding result by the equation
(prpau=(pryy-n
This must by no means be considered as an identical equa-
tion; for the first side is merely an abridged expression of
certain functional operations to be performed, while the second
is a compendious method of denoting the expanded result of
these operations. In fact these expressions will not generally
be equivalent unless @(w) and y/(w) are both distributive and
commutative with each other.
121. Cor. 3. Tf W(u)=S,.4,-1-@n-1(u)+x,.(%), where
Xn(w)=0 for some value of m and for all succeeding values ;
then we may put
WW) =SnIn—1-Pm—1(%)> (13);
44
and, if @,(w), p(u),--- Pale) are distributive functions, and
commutative both with each other, and with any constant factor,
we shall have
V (0) =(Snin-a-Pn-1)" (120).
122. Cor. 4. If, in the same case, a,,_,=a"~', and
Qn-1=g" *, then
Vw)=Saa™ gp" yu
1 - ‘
= es u, (12) and (13);
=(1-a.g)™".u.
123. It will be readily seen that the preceding theorems
of this chapter will hold not only when @ and yy are symbols
denoting functions of which the successive orders are deduced
by a series of substitutions, but also when they denote functions
of which the successive orders are deduced by performing a
series of operations all of which are subject to any given law.
An exception, however, must be made with respect to the theorem
pr. grwau:
for, if @, denotes an operation such that
p.(u+a)=,(u),
where @ is independent of w; then
db: '-p,(u)=u+e,,
Qi: Pe (U)=h,(U) +0,
Pe * pe (uy=ut hs (4) +e
n
and. de”. Pz' (u) =U+9n, Dro eM) 0.
where ¢,, is some quantity independent of #, and is to be de-
termined by the conditions of the problem.
45
124. If wis a function of any number of quantities, two
of which are # and y; then @,(wv) may be used to denote the
result of an operation in which w only undergoes a change,
and @,(wz) a result similarly obtained on the supposition that
y is the only variable; while p,.,(w) will denote @,. }@y-(w)}.
125. The symbols (w),_,, and @,_,(«) denote respectively
the values of wu, and p.(w), when # is put equal to a; this
substitution, in the latter case, not being made till after the
operation indicated by @, has been performed.
126. Definition. If in uw, any function of x, we sub-
stitute a+h for v, w will, in general, assume a new value,
which is called the New State (Etat) of u taken with respect
to x, and is denoted by the symbol E(w).
127. Definition. The excess of the new value of w above
its original value is called the Difference of wu taken with re-
spect to x, and is denoted by the symbol D,(w).
128. Cor.1. D,(u)=E,(u)—u, E,(u)=u+D,(u), and
u=E,(u)—D,(u).
129. Cor. 2. If w=(~), and #+h is substituted for «,
we shall get E,(uv)=P(#+h), and D,(u)=h(w+h)—(#) ; or,
since h is the difference between the two values of «,
E(u)=9(v+Dz2), and D,(u)=p(v+Dex)-P(2).
The case that most commonly occurs being that in which
Dwx=1, we shall denote the difference of « with respect to a,
on this supposition, by the symbol A,(2); that is
As(u)=(0+1)-G(2).
130. (Cor. 3.) DD) =D"*"(u), and D,(u)=4u;
also E,” EB,’ (u)=E,"*" (u),
Eo (u)=u, and) Ea". BA (u)=u: (110), (111), and (112).
46
131. Theorem. E,'.p(«)=$(w+nDa).
For, E,@(«)=$(v+Da2).
EP (2) =E,. E,.p(«), (110)
=EK,.o(v+Dz2)
=(#7+2Dz2).
&e. = &e.
E,'p(@)=9(@+nD2).
132. Theorem. E,-'(w)=(«-Da).
For, put E,"f(w)=Py(#) ; then
p(a)=E,.E,'p(x), (130) ;
=E,.pv(2)
=v («+Dz2).
-. 2=V(w+De), and
v-Du=\(2) ;
. Ep.p(@)=$(e-Da).
133. Cor. E,-"@(«)=$(w—-n Da).
134. Theorem. E,*\(u+v)=E,;"(w)+E,*'(v).
For, put w=(#), and v=(a) ;
then E,*!(w+v)=E,"' $(a) +(a)}
i eee
= Ba) +Ee'y(x)
=F," (u)+£." (0).
47
135. Theorem. E,*'(aw)=a.E,*'(u); @ being indepen-
dent of 2.
For, put w=9(2) ;
then E,*'(au)=E,*'. §a.g(a)}
=a.o(#+De)
=a.E,*'$(«)
=a Be (a).
136. Theorem. D,(u+v)=D,(u)+D,(v).
For, put w=(2), and v=\(c);
then D,(w+v)=D,{p(2)+W(2)}
= (+ Dx) +y(v4+De)—9(2) (2)
=(0+ Da) -(2)+W(v+ D2) (2)
=D,.$(«)+D,.V(a)
=D,(u)+D,(v).
137. Theorem. D,(au)=a.D,(u); a being independent
of a.
For, put w=(2) ;
then D,(aw)=D,{a.g(2)}
=a.(v+Dx)-a.(2)
=a\p(v+Da)-9(2)}
=a.D,.$(«)
=a.D,(2).
48
138. Theorem. D,-".D,"(u) = a Dm) te
For, let a be any quantity independent of #, and put
U=P(w) 5
then D,(u+a)=}(7+Da)+al —}(x) +a}
=(«7+ D2) -(2)
=D,(u).
De®. Di(U)=U+S_-De- (Gy) (1238)
n
=bt+ Snes if?) 1)» , 137):
139. Cor.1. D,-'(u+v)=D,-'$D,.D,-\(u)+D,.D,'(v)},
(128) ;
=D, /D,4D, W)+ De @) it. a6);
=D, (u)+D,-'(v), (188).
The arbitrary constant must be added after the performance of
the operations indicated in the second member of the equation.
140.- Cor. 2. D,-!(au)=D,71fa.D.D,-"(u)t, (130) ;
=D,1.D,\4.D, "(u)t, G37);
=. 1, (a), (138).
141. Theorem. E,.D,(u)=D,.E,(u).
For, D,(u)=L,(u)—u.
+. E,D,(u)=E£,E,(u)-E,(u), (134)
49
149. Cor. E,-D,(u)=D," E(u),
D,E,-"(u)=E,~'.D,(u),
and, A? D, (y= D>. 2, * (uy
143. It follows from the last nine Articles that the func-
tions denoted by the symbols H,*", D,, are distributive, and
commutative with each other and with any factor independent
of x.
n+1 w
144. Theorem. D,"(w)=S,,(-1)"7 ea Ee" (u).
n—
. For, D,(w)=E,(u)—u
=(E,-1).u, (115).
~. Df (u)=(E,-1),4, (116);
nv
n+1
=S,,(—1)"-). =. H2-"*1(u), (119) & (143).
[mat
n
n+1
145. Theorem. E,(u)=S, “=~. D,""'(u).
[moe
Hor) EB ¢a yas DCs}
=(1+D,)u, (115);
E2(w)=(+D,),u, (116);
Nh
n+l
=S, a .D,"""(w), (119) and (148).
[m—1
G
50°
146. Theorem. D,'.v"=|n-h' ; where h= Di.
Yor, D, .2"=(a+h)’-2"=n.2"~'.h+inferior positive powers of «#
D,’ .2"=\n.v"~*.h’ +inferior positive powers of «.
9
4
D,".«"=|n.a"~" hk" +inferior positive powers of 2.
™m
DEG Sh
147. When the quantity of which either the difference or
new state is to be taken is a power of the independent variable,
the index subscript of the letters D, A, or E may be omitted ;
and hence the above theorem will be expressed thus:
D".x"=|n.h".
n
n+1 s
148. Theorem. r"=$,(-1)"). = —7 (w+n—7r+1)”.
n+l [n
For, D’ nm =S, €—1)' fay a =: ee r+ a", (144).
at+l
. Nt GS, (1). r—1 -(w+n—-7+1)", (131).
atl
149. Cor.1. S,(-1)'"?. —*. (@+n—r+1)"=A".a", (148) 5
[7-1
=|» (147).
a+l | %
150. Cor.2. A*.0"=9;¢=1)'3! feat (—7+8)".
The numbers comprehended under the symbol A".0” are of
great utility in the expansion of various functions. The following
values may be readily calculated by the theorem of this article :
OO8839S | ODJGSEOL | OOOOFGOE | OOSSEQ6S | OFFSSFOL | OOOSOTS OossIs
O8869& OGSISFI OSF86S6 OSI G06L O8t9O8I Og I8I
OSIIFI OOO9SI POSOP 96LG
———<—<—_— —$—$—q—| Kqe (mm — ———
OFOS OOsgT OOFS8 9081
OO8T O9ST OFS
— Oot OFS
52
151. Theorem. _D,*.a"=a*.(a'—1)’;
For, D,.a’=a°t"—q*
=a" (a"-1)
D,2.a*=(a"—1).D,.a", (137);
=(a"—-1)’.a*;
and, similarly, D,”.a*=a*.(a"—1)".
152. Theorem. D,.\a+ba=bnh.|a+b.(@+h).
n, bh n—l, bh
For, D,. a+ba=| a+b.(a+h)—|a+be
n, bh n, bh n, bh
= }a+b(w+nh)—(a+b2)t |a+b.(w+h), (41);
n—l, bh
=bnh.|a+b.(a+h).
n—l, bh
153. A,.|w=n.| a.
n n—]
S22 2 2 aah
n n n
=(v+1)|e-|x.(@—-n+1), (41);
m—l] n—]
=n.| wv.
n—J
1 —bnh
154. Theorem. D,—— =
|a+be a+ber
n, bh n+l, bh
1 iT ]
tate De (atiae (ach ein (aebo
n, bh ny bh n, bh
at+be a+b.(@+nh)
|a+be a+be
n+l, bh n+l, bk
156. Theorem.
DP .d{a4(r—1).b} = {G(0+nh)—G(a)} .P.d(w4rh)
For, D,.P,p{a+(r—1).ht =P, p(w+rh)—P,p §w4(r-1).h3
= [p(w inh) —G(a)} Pp (w+ rh), (82)
157. Theorem. D,.[P,@ e+ (r-1)-h >
=~ {p(v+nh)—9(0)} [Bp $03 (r—1)-h}
For, D,.[P,o{e+(r-1)-ht}
=P, o(w+rh)t [Pg fat (r-1) 42 J
be Sp(ainh)—p(2)! TP, Sv+(r-1).h} le:
m m—Ts,
158. Theorem. D,.P,(u,)=8, Coy. (t,-De-t):
For, £,..P,(a;) sp) (u,+.D,u,)
m+1 m—r+1, r—1
=S, C,.(u,.D,m), (67);
m m—Ty
ob (2, +S, C, a. Da: (9).
m
ED, Paar) — S, TORE. Di))5_(128).
159. Theorem. D,(uv)=u.D,v+D,(u).E,v.
For, E,(wv)=(u+D,u) (v+D,v)
=uv+D,(a).v+u.D,v0+D,(u). Dv
=uv+u.D,v04+ D,(u). Ev;
. Duv)=e.D,0+D,(u). Lv.
and
DIO,
1 D,(u).v-—u.D,v
160. Theorem. D, (“) = a
0 v.E,v
wn “-+- Dia. ua
For, D, z = -
v v+D,0 v
uv+D,(u).v—uv—u.D,v
a ane ene Pe
v. Ev
D,(u).v—u. D,v
if v. Ev ,
161. Theorem. If (vu), Pi), W(v), and y,(v) are
distributive functions, commutative with each other and a con-
stant factor, and if P(w).W(v)+,(w)-Wn(v) is denoted by
(pW+qd.\1) uv, then shall
n+l |
Gis ates EO @)
For, by proceeding precisely as in Art. 118, we shall find
that
(DY+ br Pa)n0= Sn eG" Pi" (W) YP" (0).
| m— i
Suppose, therefore,
”
n+1 Sih.
(ov+gi V1) nUV=Sm re 1 Gis (w) AN oe (v) ;
then
n+l |
(pw + QW) np 1Mv= (pw +i) sees ; orm pi” (w). Wee Wi" (v)
50
SE Ee OLY OO]
FAlPrr GPW] hLe" a @)]}, (13)s
= S., = Names (a0 (72) Nata a \i gms (2)
j + Gr" Lit (a) mL a” (v)?, (114)5
nN
= p""'(u). YOST ee soar) ™ (41). y" m+1 Wn" (v)
ye
is S., aa Po) dy" (U) Wot (ve) +"*? (u) Wi" (&),
(Ghee
=p" (u).""? (v)+S,, - im Oy (at) Dy
+p (w)-Wr'*"(e), (5) and (47);
By eee!
= S. eas . (Og maie Cen (w) , Neen ; ee (v), (9).
If, therefore, the law were true for 2 it would be true for
m+1; but it is true for 3, and therefore it is true for 7.
162. Cor. 1. The equation just found may be written
thus,
i
S iz
(ov+gy Wi) Uv= Say fog m+1" aM ial" ae,
or (@+gith).uv=(+givy)’*uv. See Art. 120.
It must be carefully observed that, in the expansion in-
dicated by this last expression, the symbols @ and @, are to be
prefixed to uw, while x and \, are to be prefixed to v.
57
163. Cor. 2. If di, ps---py, and Yr, Yos...,, denote
distributive functions, and commutative both with each other,
and with a constant factor; then shall
(SP) wv=(S,Pv,)" wr.
164. Cor. 3. With the same limitations,
(Br Oe Ds. Dy) nU- Uae» Us=(Sp Pr Pree Pr)” Uy Une» Us 3
where the symbols ‘Dis "Prise Drs are to be prefixed to 1, Uo5.--Uss
respectively.
165. Theorem.
n+1 =
De (u D) S,.-—— imei m—1 Wey = (2) p De —m+1 y JD ke —] (v).
For, D,.(uv)=u.D,v+D,(u).E,(v), (159) 5
=(D,+'D,.E,)uv, (161); where 'D, only belongs to w.
* D2 (uv)=(D,4+'D,Ez),-U0
n+1 |
= ee DS a) De et (oy (On):
| m =|
166. Theorem.
n+l n
De? (uv)= Ss. (-1)"" 1 ET sl Es ml). nae mato):
For, D,(uv)=E,(u). E,(v)-uv
=(KH,.E,-1)uv, (161);
. D2 (ur)=(L,. E,-1),.Uv
n+1 nm
=S,(-1)""! CT sees 8 ()..-* 21a), (161).
H
58
167. Cor.1. D,"(wu2)=$(+'D,)(14+°D,)-1f "wu 5 and
DE. P, (w,)=$(1+'D,)04°D,)...+"D,)-1 i gd
= 1S. CCD)" Uj Us.+.Uny (68) and (9).
168. Theorem. E,.E,(u)=E,.E,(u).
For, put w= («, y); then
E,(w)=(«, y+ Dy)
E,.E, (w= («+ De, y+Dy)
=F,p(v+Da, y)
=E,.E,.p(«, 9)
Sr Oy
Hence, we may express either H,.E,(«) or E,.E,(«), by
E,,,(u); while D,,,(«) will denote E,, ,,(w)—w.
By
169. Theorem. E,E,(u)=u+D,(u)+D,(u)+D,D,(u).
For, E,(u)=u+D,(u), (128).
E,. E,(u)=E,(u)+£,-D,(u), (134);
=u+D,(u)+D,(u)+D,D,(u), (128).
170. Cor. 1. Since w+D,(1)+D,(w)+D,D,(u)=E,E,(u)
=E,E,(u), (168) ;
=u+D,(u)+D,(u)+D,D,(w).
-. DID (o)=D, PGs):
171. Con.2. £,,(u)=(14+D,4:D,+D,D,)u, (115);
ob” (u)=(0+D,4+D,+D,D,),u
“ay
=(1+D,+D,+D,D,)".u, (121) and (143).
59
172. Cor. 3. D,,(u)=(D,4+ D,+D,D,)u, (168)
a Dy (w= (D,+D,+D,Dy,),u
=(D,+D,+D,D,)".u, (121) and (143) ;
= 1(1+D,)(14D,)=14"
nr
n+1
173. Theorem. D,’,(u)=S,(-1)"7 Tee i eaten (¢))
ry
For, D,,,(u)=E,,,(u)—u, (168);
=(£,,,—1)u.
. De é)=(Esy-Y at
=(E,,,-1)’u, (121) and (143);
n+ a
=S,,(—1)""? = a Oi Py c
a
n+1
174. Theorem. E,”,(u)= Sn—_— Dr y (%)-
[mai
For, E,,,(u)=u+D,,,(u), (168);
== (VEE DRS ore
Ex, (@)=GA+Dzy)a¥
=(14+D,,,)".u, (4121) and (143) ;
nN
n+1
So Pr 0)
60
175. Theorem.
Lm [Ps
He _E}(u)=8, 8, DF Ds ae:
Pas pat
|m
———.D,"~\(u), (145) and (13).
aa
For, £,"(u)= S
| m
E.”. Ej (w=S,-—.D,""!. E,"(u), (168), (134), (135) & (141);
Pend
nv
aS, = D8. ——.D,’~'(w), (145) and (13) ;
=§,-=.S, .D,'-!.D,5-}(u), (136) and (137);
=S,6,-2 = D-5D,!-1(u), (6) and (18).
176. Theorem.
D,-) (uv)=u.D,-u-D,'$D,(u).D, "Et.
For, D,(u.D,~'v)=uv+ D(u).D,~'.E,(v), (159) and (142).
. u.D,-v=D,"'(uv)+D,-'3D,(u).D,-1E,(v)t, » (139) 5
and D,-'(wv)=u.D,-'v-D,-"}D,(u).D,"".E,vt.
61
177. Theorem.
De? (wv) = S,, (- Lye it Dent (w) \ VD me Ee -lay
+(-1)".D,"! De (w) 1) Le Oaks vi
For, Dis 1 {DS =) (w) : DD (m—1) pee =i v i = Diz =| (w) : yD ig JD —] (v)
a Dis 1 ; Diz (w) ‘ Dae ; y Die v ¢ . (176) 5
: (- 1)" me Die! ; De (w) } Dix (m—1) ng Een ;
Ne (-1)""" ’ Des 1 (w) ; Dee BY Oe 1 v
=s ( al i ; D7 DS (w) ; Das ; pe v :
o) S,(-1)"7 : DH 0 ae (w) oD ea 1) : E,""'v}
a Ss. (- 1) -] ’ Di -1 (2) 5 yO us ’ yOu -1 u
+S,, (- L)* ae Ds (w) : Dw E,"v} F (5) ; :
r—l
D,~) (uv) +8, (-1)". D7 {D2 (u).D, " Ev t
a S., ( pms 1 -1 ‘ De -1 (w) P Dp m Y Bec -1 a
r—1 “
mn S,, ( fies ine ’ Dp 1D? (w) ; De ) Dee vi
(=). De? } DD," (u) WO 1 ORE 3 (9)5
aieh Ds 1 (u v) =§,,(- ie \ Dis 1 (w) _ D-* DB rh
+(-1)’.D,“' Ay, (w) Di hut !
178. Cor. If, for some value of r, D,’u=0, then
Dix (u v) =S,,(- Her! j 1D are , Dm Pee, (13) ;
= {D4 (14D. 2D, * Eee;
where 'D, only belongs to wu, (12) 5
62
D,-"(wv)=$D_-".(14'D,.D,1.E,)-"} wo, (163) ;
n
Dp: —n 1S,(=1)"- (ima at oD: m—1 SD —(m—1) A pers we,
(92) and (39) ;
i |
=S (-1 m—1 _m—),1 D m—1 D —(n+m-—1) m—1p,
m ) . 7 . v (w) 5 r 5) oe v.
in
179. Theorem. D,~'(a* Jalen 7, + const.
For, a*.(a’—1)=D,.a*, . (151),
v
a
wy be 137);
Slagle (137)
a
anda 3) a, ; teonst. (138).
a
nL
180) \ (Cor. °\0D.-* fae" =1) 5", eae eat
181. Theorem.
D> '(a,.) =O 5, (= 1) a Ge 1) Dn
4(-1)".D,-' 4 Ds (u).a"**. (a =1)*t.
For, Di. (aw) ae (21)? D2) De oe
+(—1) D3 Dy) 2D Eso KV
qtt (mh
=$,,(-1)” *.———— :
(ayn
(180) ;
Pe
=a" .S,(—1)""} ain DF, (a* 1)" DP
q7ttrh
.D2""'u+(-1)" D-\p, (zw). (a =}
(1). DOU Day." (PAD, CY:
LSE is abe =, a+b(a—h).
eae bh(n+1) bantu)
For, bh(n+1).|a+b@ = D,.|a+b.(w—h), (152)
n,bh n+1, bh
|a+bwe = D,. \honn a+b(a 1) (37)
nibh bh(n +1) 7 Oa
and DD," a-ba=————— | |a+b.(#—h).
eee bh(n+1) n+l, bh aa
v
PS Bee Nee 2 oo
ao Geel
(n+1) 1h | w (153)
n n+]
av
so jv=A,—, (137); and
L— n+1
ja
NS, e= nl
le n+1
1 1
Nga YDS 8 ee ee
|a+be bh(m—-1).|a+be
ny bh n—1, bh
—bh(n-1)
For, ————— = D,.——— 154).
i ja+ba zi jat+ba’ GC)
n,bh n—1,bh
1 1
——— =D,.)-—__—_—_ 137);
G+ bay tf | hasta: ( )
n, bh n=l, dh
1
and D, oe, ea ot Da
ieee bh(m—1).|a+be
1, bh m—l, bh
64
—1
m
iad ce (n— (n-1)|@=1
n—-l
_~(n-1
For, Me ) Seems (155)
[a |w—1
ee n—l
1 1
- US7)'5
|v 4 (n—1) =} Ce
nr 7
1 -—1
and 1 =
nd, v (n—1)|v-1
r+
186. .A-.7=S 7A" 6:: = » being any positive integer.
+l [2
For, (y+«)"=Sn—--A.”"'y", (145).
[m=1
eo |v
o, #=S,, ———-A”-!.0";
fal ee
1 ee zat Nie 1 0” if 3
and As ae a |e, (139), and (140);
ya m—1
r+ v
aS. (Amaiigs en (ass).
[me
187. Theorem. Sy tee Nees a) nsas
n n+l n
For, A Tie Nee an, = ra Srna Am —Sn Ay,
=An+1:
and
or, as it may be
conveniently expressed,
n
oy An= (A, ; =A) Gn+1-
CHAPTER VI.
ON DIFFERENTIATION IN GENERAL.
188. Derinition. The quantity }(Dx)-"D,"u}p,-) is
called the n™ differential coefficient of u, taken with respect
to x, and is denoted by the symbol d,"w.
189. Theorem. d,"d,?u=d,"*"u.
For, put Dw=h; then
d7u=(h*. DP);
and -d7" 0 u= ha" D2 (hk "DS U)
Rho De (he as,
=3h it” Deal, 2.) (187) and. (180) 5
=A? a.
190. Cor. dQu=w.
191. Theorem. d,.(u+v)=d,u+d,v.
For, d,.(w+v)=$h-'.D,(utv)t <9
=(h-).D,w+h7 .D,e),5 (36)s
=(h-’. Du), 5+ (hk .D,2)),<9
=d,u+d,v.
67
192. Theorem. d,(w+a)=d,w; where a is independent
of 2.
For, d,.(w+a)= $h-*.D,(u+a) t ro
=(". Diu), -)0 2138);
=U:
19S) eC ors ty d,.¢—0, 7 (198):
n
194. Cor. 2. d,-".du=u+S,d,"-™ «Cm Where c, is
independent of #, (123).
195. The symbol f,’w is equivalent to d,-"w, and is read
the n™ integral of u, taken with respect to x. Hence the equa-
tion just found may be written thus:
n
f?d,"u=u+S,, hope: Cin
196. Theorem. d,(au)=ad,u.
For, d,.(au)={h-'.D,(au)},
=) hia. Du. = (137)3
197. Theorem. d,.v=1.
For, d,.7=(2).Da),,
=(1);,<0
=;
68
© h-
i " é = aia a,” wv).
198. Theorem. o(w+h)=S Sip p(x)
For, put Dr=k, and h=nk; then
n
p(wt+nk)=S,, = -D,"~*.p(a), (145).
nko }41
- P(e+h)=S,, ESS Pik a
whatever the value of k may be;
Se hk-?
LGR m—l n—l
S: |m—1 = D; ; p(2) , k=0
xs h™-}
— ie lm aa il ‘ Dey i; p(x) a
=S,,—— .d,""1. (a).
o hm -l
199. Cor.1. H,u=S,,— -d"-u,
ca
x m
and D,u=S,,—:
[”
200. Cor.2. D,uw=(e'—1)u, (106) and (115).
dU.
< DP u= (eh —1)"u, (121).
69
: ; t h™ 5
201. Cor. 3. d,”".p(x) is the coefficient of — in the
|m
expansion of @(#+A).
202. Theorem. If wis a function of v, then
d,.~(u)=d,.p(u).d,u.
eee) Hearn (27
For, D-H) BE (D.0) (199) ;
x a" : plu) co he
= ; a pee m 2
Sn i oe (S, poe wu)", (199)
ad (aii ee) a (S.ar . ia) is (6) ;
“and d,.~(w= Sh-’. D,. pu) ; h=0
=d,.p(u).d,u.
203. Theorem. d,.(wv)=vd,u+ud,v.
For, D,(uwv)=(u+ D,u)(v+D,v) —uv
=v.D,u+u.D,v+D,(u).D,v.
ho. D,(uv)=0.h7.D,u+u.h-.D,0+h".D,(u).D,v;
and {h7!.D,(wv)},29=0- {h7'.D,u) 9+ Uh. D,0} 5-9
+[{h-?. Diu} ,29-(De)r<o OF COPE) ar Momaed BRO ||
o. d,.(uv)=vd,ut+ud,vt {d,u.0 or 0.d,0}
=vd,u+ud,v.
Ah Cain, — =a
d,.(wv) d,u d,v
: f —— -——,
Uv ul v
d,.P,u, ™ d,w
205. Theorem. ———=S,, a =
P..u, z
"P. P
is sehgl hy ae Ges beag tla Th
For, a — =", a scully) (204).
Pu Pu Un+)
Yr Y if i:
n
d,. Poa, dU i A Uns+1
oS = SLs Sn——> (24);
n
U u
iP: Uy 1 m+1
dU,
=p, — 5) (9).
nsm
1 na
206. Cor. d,.P,w,=S,,d,tm-P,u,-
207. Theorem. d,.u"=nu'—d,u, for every rational
value of .
G20P 2a
For, — =S, So (205).
P,u,
dow * du du
Put w,=u, then =S, =o).
T) He U U
ws 1d, wan -* dyes
nm being a positive integer.
Also, since 1=2".w-",
O=u-".nu"~"d,u+u".d,.u-", (193) and (203);
and d,.w-"=—nu-"—d,u.
it
+
. +) . are °
Again, uw "=(w ")"; mand n being any positive integers.
m
m
tmu~"~'d,u=n. (uw oy eR (u ”)
+mzm = 5 Ese
=nU "d,.(u ”)
+” m Le
andsd..(% “\—=-— 2 * deu
n
Hence, d,.u’=nu"-'d,u, for every rational value of 2.
208. Theorem. d,. (=) ber ( dU =").
e v\u v
Kor, diz (=) =d,.(uv—')
Uv
=v~'.d,u+u.(-1)v-"d,v, (203), and (207);
u (du =")
(= vo)
m m mm
”
P,u, Pw id.Pu, .d,.P.0v,
2
209; Cox. 4,4 a |Wore yee
P,v, Pio" (Piz, P,v,
21 0. Theorem. de ae a=l|n-a"-™.
m
For, d,.2°=na"—', (207) and (197);
d,.@=n.dia'=*, . (196);
—2.
=|n.a" >
2
&c.=&e.
I Mm mt _ sn—m
d,” a =|m.0
m
Lea 5
12
n
, n+1 |
t — % :
211. Theorem. d,’.(uwv)=S,——.d,"""*1u.d,""'v.
|m—1
For, d,.(wv)=vd,u+ud,v, (203);
=(d,+'d,)wv, where 'd, belongs tov, (161).
.. d,".(uv)=(d,+'d,),Uv
=(d,+'d,)’uv, (162);
n+1 |
epi n—m+1 m—1p,
=, ———.d, Uu a, Bis
| m— 1
2 A"! ce) hm} \
E,(u = (S. aul : a!) ae a EE 0) ’ (199) ;
ra) n d N—May m—lpy
=S2" 7S, = = sm (26).
m ‘ihki> 7 acall aeee
| 7 —m |m Ly,
d.”. UU n+l Ge de Ow
gh aD Ape ar oe
im |j2—m-+1 |m—I
n+1 |
and d,”.(wv)= Ga dt de.
|m—1
m
912. Cor. d,.P,u,=(Cd,+7d,+--.4+7d,)U,Up.--Um, (206).
m
ody". P,w,=(Cd, 47d, +... +7dz)"UyUne0-Um, (164);
m m
=(S,’"d,)".P,u,, where "d, belongs to w,.
73
213. Theorem. If w is such a function of w as may
be expanded in positive and integral powers of w, then shall
For, assume w=S,,a,_,.v"~', where a,_, is either zero or
some finite quantity;
then d,"w=S,a,_,-|m—-1-0"-""1, (191), (196), and (210);
m
mm foo}
=5 Gq —1-[M— 1.0") 4 | MES, Ant n m+n.x", (9);
™ m
lo =)
. —" |
=U. | m+ Lae | m+n.a", since | n—1=0, (°=)).
m ™m
1
re m me m=1
: Os U=Ay,.| My and a,,=—-d)_ us (
‘m M= OF
1
a | =e
Also (u),_,=43; and therefore, a,,_1= (maa = OD CEA
© ym}
q m—1
and pera at, - 0:
214. Theorem. d,d,u=d,d,u; x and y being independent
of each other.
For, put Dw=h, and Dy=k; then
dju=(k-'. Dy w);,-o5
and d,d,u= $h-'..D,(k-'D,u),-9¢ n=
=$h-*.(k-. D,D,u) p43 95 (187);
= ik DD ees
=e A}. D, Du} 5», ae sh 170):
aah) DD, (ho. Daye - as CS7h3
=a iat DD, (hc Sanya |
=, 1D, dur»
=d,d,U.
K
74
215. Cor. di,” : dU ot a, f d,” U.-
216. Theorem.
m—1 Jpn—1
h Mat lay |) ‘p(@, y)-
@(w+h, y+k)=Sn 8, ema
© Am)
For, p(#@+h, y)=Sn—— -4,""!.p(a, 9), (198)5
|m—1
io) m—1
and p(w+h, es Apt d,""'. h(a, y+k)
<=) he 1
= Bi Ra 1 a 1 y ; 198 :
Dn |m—1 tea ae D(a, y) ( )
roa) A@-1 f-}
: ge gO) Ca) (191).
ioe ‘d,'" D(a, ¥)s
(196), and (6).
Bien) Wor:
he Mt) fn 1
p(wth, y+k)= Sa "|m—n|n=1 as i Ras i lm : -p(2, Y)s (18).
218. Theorem. If w is such a function of w and y,
that it may be expanded in positive integral powers of «,
and y, then shall
GET aly ae,
[==] fo=) <
= : -1 - .
For, assume, wz=S,,0 5" 29,9" 7G nas where @, jas
is not infinite;
daw =
y=0 a = i.
then s—1 ae man »Un— 1, s=19 (zap (213) ?
du Ps
y=0 ai eT n=1
= 5,0" *OUn,-1, n-19 Cos)
Sie ERY
a d a)
And, se aan eet (Ce eae C2 E3)5
1 4
dr 0 d/o Ut 0 m=1 Cx a
> n=coJ ?
oh [m—i.|n—1_ =Qn-1, n-19 Gee,
o ym Ute
F *s m—1 n—1
and w=S,, eee a re c dio Gi U-
219. Theorem. If x is a function of & and y,
then shall d,}d,z.@(z){=d, Jd,z.p(z)}
For, d,.d,} f.p(z)} =d,.4,) Lp(z)}, (214);
d,}d,z.d,. [.p(z)t =d,}d,2.d,.,p(z)}, (202)s
or delays sp(s)}=dy}dex.p(@)}
220. If y=vW$x+a.p(y)}, where x is independent of
w and y; then shall
SM=fV +S, ao d."'S p(x) |"-d.- fy (a)}-
Bor, f)=1FO) }x-0 Sa
ds f(y)> (218).
Put x+a.o(y)=u;
ad Paper
then — bs d, Wy (2)
d.y d...y(w)
_ d,w.d, yu)
— d.u.d, (uy)?
du
dat
—pty)te-d,- py)
eT eee d..p(y) i
(202);
76
. Sis¢a.d..oy)idy=ip(y)+«-d,.py) dy,
and, cancelling identical terms,
diy _ dy- PY)
dy d..p(y)’
Also, d,-f(y)=dey f(y)» (202);
=d.y.p(y)-d,-f(y)s
d,’. f(y)=d,\d.y.p(y)-d,-f(y)}
=d.}d,y.o(y)-d,. f(y), (219);
=d.}d.y.p(y)|’-dy-f(y)};
and, similarly, d,”".f(y)=d.""'.4d.y-¢(y)|"-d,.f@)$
=d.""!.Sp(y)|".d.- f(y}, (202).
of Mad ' {PV@)]" 4 SV),
d,y=d.y. p(y), since (202).
ve
weal FG) “MO)+Sa, dG) de AR)
221. Cor. If y=z+a.p(y),
a
then f(y)=f(2)+Sn im a 5(s)|".d,.f(8)}-
222. T'heorem.
n <
£(@»r)=S, (-1) da 4 ferv+(-1)". f Sd,"U. ftv f
For, ip Noh men | Bie ‘yt = qm" Te v— if NORE ibs vt, (203).
(=) eee” 7 te. | eer
=(—1)” sei R a v+(-1)".f, Sd,"u. fv?
i
7
Sa(—ayet fi fde'n. f2-10}
n “
= se(=1)" > dt. [048m (—1)". f, Sd,” Uw. [eor, (4) and (5):
n—1
f(uv)+S,(-1)"f.sd,u. fret
n n—I
=S,.(-1)"71.d."—'w. ["0+S,, (- ye S di,” U Bo vt
+(-1) f,$d,"u.frv', (9).
n
4. fo) =S,(-1)"-1d,"-1 0. v4 (-1)". Lf dru. fr0t.
29S: Cor:
n
v
n a”
frun$,(-1 deus EL de, (197) & (210).
CHAPTER VII.
ON POLYNOMIALS.
Msn
994. Tue symbol §,,,,('a,) denotes the sum of every
term that can be formed with the following conditions: each
term is the product of m quantities in which 7 has the values
of the successive natural numbers, while s has any m values
such that their sum shall be », zero being admissible as a
value of s, and repetitions of the same value of that letter
being allowed in the same term. Thus:
3,4
ql 2 9
S,, 45 Fa,) =*0, Ay°A +70, 'A2°A3+°Q, dy 'A3+"G,"Ae"A3+
26) gg) On ig 10, On Oat Oy By Gt
Ne 570g ls gd, le in On la
OG, Gn Az + °C; oO, + Oy Gy 1O-
Msn n+l m—l, t1
BO. COR.) Seta =O Gana): &
m eke a 1 es
226. Theorem. Co ae
n is
] 2 a+l a n—t+l Ge
For, ) =3(5'2)'=s:— “ (88) ;
it . > t ’ ?
| ra) |~—t+1 |#-1
2,n a s
79
Suppose, therefore,
m—1 m—1,” §
aN n X Ivey
In ; (S,4,) '=S,, +s]
c 5
1 m—1
then —.(S,a,)"= Ty (4,,+8,a,)"
n+l } n—t+l1 m—1
=a us Say) (88):
‘|m—t+1.|t=1 C2) Ce,
n+l] awa) eels tly By
a7 ‘\n-t+1 ¢ miu
m,n he
=$,,..7—> (225).
[s
If, therefore, the law were true for m—1, it would be true for
m; but it 7s true for 2, and therefore for m.
1 m m,n ads U,.
. d," . P,w,=S,,.;—— ’ (
" c
I
227. Cor. 21a).
228. ‘The symbol Dw" p(a) denotes the coefficient of 7”
in the development of @(S,,@n_:0”"~'), which coefficient may be
called the m™ polynomial coefficient of P(a@) taken with respect
to a. In this symbol the index subscript of @ is the letter
according to the indices subscript of which the different powers
of # ascend, and the quantity following the functional symbol
is the term independent of & in the series S,,a,,_;a"~!*. If the
index subscript of @ is omitted, that letter is understood which
* Throughout this Chapter a is put for a,, for the sake of brevity.
80
immediately follows it, and if the function is a power of the
polynomial, the parentheses including the first term of the
polynomial may be omitted: thus
@”" (a) denotes the coefficient of x” in S,, dn-1(a) a",
wm” a’
28
m—1 in
mm — 0 § ?
m m m—len
Dm a, EPR sac 355 SA ee A ei t :
229. Cor. w,’.p(a)=(a).
2 m wa” =i a
230. Theorem. @w"a"=S, | Sr Aa
pe
value of mn.
co co
For, (S,.0,.30" 7) =r ciStala ys (9) and (6) ;
~
co ce
ised era a Wa ihe (92);
foo] oe
ps ml. —m+) -1 _— y— =
_ S2 =a at mt) aml, S,.2’ 1. a La u (228) :
[a
m
ters . qn-mt+t a" 1 ‘ 0, a (6) and (18).
pad m —
aa SE a
m+1 | fe
a" a"=S, m—r+1 qu-mt+r-1 wow Ta (228) ;
>) >
m—-r+i
|
m
=5, tl at lay"), since @”.a,°=0;
|m—r+1
m | 2
= pet eg ae (8).
["
931. Corl: @af=S,—-6/"o "G,))-
232. Cor. 2. If is a positive integer,
Y
m n m n—Y m—r
Dm .a, a, -D a, 1
— =p, , and
fae
ow". a. m Geog stars : ow! ae
| 7 ce er m—r+1
233. From this last value we may deduce any number of
one a"
terms of the expansion of Ee , much more readily than from
n :
the general expression for that expansion. We have
wo” .a” q-™ w. ay m quamti ow = m—1
| | —m- ae * |n—m+1 —m+1 | m es
qzu—m +2 : D. Ge
+ |~—m+2 . |m—2
Hence, putting m=1, and n=m—1,
+&c.
= 5 9
a” m+2 SGN Gs
|~—m-+2 . m—2
And, putting m=2, and n=m—2,
= —2
m a” m+ at
mw” a” a —m ay,
a” m+2 a m—4 , Get ee
oe + . a;
|z—m+2 m—4 \2 |m—3
qr-mt+3 : wD .a m—-3
82
And, proceeding in the same manner, we shall obtain suc-
cessively the following terms :
My? Gan er Oo —m+1 ei a, m—2
ine y |n—m |m —m * [n—m+1 |m—2 m+ | m =50
qm +2 Ce t os a” 3 qr-m+3 re : ie
4 |m—m-+2 he a: era m eee |3
m—s5 a a? m+4 ay m—8 RU he peas F fies
|m—5 m= a + | 2— imoneeen 8 |4 * m=7 Sic
| Gage a 2 Com qramts a2. Gs
“ |im—6 (i +4) Tm af 33 Ae m—10 |5
+ EES 3+ ie (« a at a a ) i iia (d34,+24;)
|m—9 3 Posts} m—8 we [2 Sad |m-7 34 12 25
Ce ae " qr-mt6 joe Ga 11 sg
"6 Ae |~— —m-+6 ||m—12 |6 |m—11 |4 [a °
(Barmy ase : Ay a ea a; 3 i
joao (ie [2 *Te*) — 5 (fa toes am
io ie seg quamti Gots - rp
ar as (E +430; bast) + l\m—7- a + Sree aa [7
‘ Ce ae p50 a, —12 i a; a)
mata [5° ** wie \Ys 2 *
ai” =11 a. age as
ap ES Co + lor » A, A+ [3 <as)
a? The! (e Sic E ee 33 ay p
m—10 2 0 4 2 l2 3 | |2 )
qz-mts Gat ag qa : as
*Tn=m+s m—16 |8 an ane 6 -
age; i
ages ae ah
+ [mai ISIE. ~ [es eae [a,a,+a34,]+ E 4
= 2 n—9
Ci sie 10 ie Ga
1
——— | — +. @, ig + 437+ Gg) + ——— - @gp+ &e.
a \(2 Bo sagt i :) | m—9 ,
Whence, giving to m the values of the successive natural
numbers,
wD: n q”-1
SS -
| 2 |~—1
mw 4 a” q'-2 : Oar a?!
= + As
| |n-2 [2 [2-1
DZ } a” a’ 3 : a, q’-? a}
= Se qe AN ear ~
| 7 |n—3 [3 |n—2 [2-1
‘ 2
An A,+-@ a ( +
|n—2 ° is iti [2-1
|
Cee ee ee ea 7 + G0 eee ela
‘ho Pa eT
a -—3 as "ie a ag
+ E TUE a] Heh a + |n—2 ef era
N-3 z 2)
aq" }
2 au A:
OR ue x, COR Nee gl Le OS le 5 kan
Ce Cea Ceca CC
q’-4 as? aye O° a’-3 Tie
+ mae - Leet | + male”
aa ; at ane
+a (et a.) 5 l2 a} cy |n ; p+ My A+ Us
a’
je
o.a a *.as a '.ay Cara, Pe
(EEE
|n=5 [Sirah
oe ee (hy om be
+ Tee 1 +434, }+- a
ale eet Ble te) sy
an}
— 5 +s + As ae aes
n—I
Cir |
Salk pai (ttt) je.
85
1 Sm (ag. 5,07 *)
[o=)
ee ee n being
234. Since
any positive integer, (226);
s
Grad ys (=1),
E
Capt a” on ar : :
E =the sum of all those terms of §,,,,—— in which
n
Is
s(v7-1)=m.
c,n
=9e46
This equation may be thus written:
AY
wm”. qt 2:n,m a
‘r—1
[7 =Or,+s,4+s(r—-1) [=
m — -
235. Theorem. w”.av'=a—'.A,, (=) :
a
co
<=)
Hor, (SG, 12) na yer 9) a) (228)
o f=]
2c) iz (S,, Bin 0") (S,a"-? wo’ 1 Gas)
oe m
SS et Ope clo mn Ame (18);
© m+1
Seed Sade. @ a) Ors (9).
m+)
a a take Se
0=5,,%,) apie a
m
1 ail =| —1 -
a SP Qm-n+1° wa a +a. a" a> (9) 2
m
-1 ®@Q a—1 7-1
=U On 2m—n41 wO a
m 6
=S,,( " erie, (8)
7
ee — (tH Cm) A..(— =
m
—a,
=0.A,,(=).
a
236. Theorem.
S a a 1 fa] m n—-1 =p
m@~m—1 F a (es = t
a Sa" she (pee) LAS, (=) 4
oh dase
r
-l co =
S Bye” a 5
For, = (5,4, 10) (O,0n ae. 4) ry subyjeet
0,1
to the condition that all the coefficients after a,_,, and b, ,
vanish;
=S, a" 1S, Onn (Ons w( 228), -and (26).
m n—1
co ” —b,\
SSS Eo PaaS Ce OO 5 As (=) 5 (235).
d n U n ao ™ a™ d. m Uu
937]. ee =S,, a," . p (w) . _|m 39 where Gy —-1> im ;
co amo
For, p(ut+ D,w)=Si—
oo iB n—l o he
ae Q ie p(w) (S;, al : aru, (199);
| mn
ec n—1 | ds M ayn n—-1 >
Jae Ly - P(t) ht 1 (s., hi'- : 7 (6) ;
nL
|n=1 ’ :
o h®- l
=S, ‘i \d*s 1 oS ae og a
n—
du
where @,,_1= > /(228)5
=) d,, n—m
=e gigas «PNY PO) i a nats (26).
— [n=m m
. d,”".b(w) ar dr-"*'d(u) Bp ke vice (201);
| 7 a: |z—m-+1
G de" ** @(u)
m EE aa
One aa amas a since w'a°=0:
|w—m-+1
n—™M pm
a
=Sud." p(w). acre io:
238. Theorem.
espe n—m+1
eo
PS, Qn-1 an 5) os (a) +S, a. =) da. ad) pl).
|2—m-+1 ;
= ee
For, PS, An—-1 a 2») =p(a+e Z S., Cy BO zy
=$(2)+ $50 G v- Sdn)", (198);
ae 2
BL suas ).-Sn2 m— 1 ao" Te (228);
m— la n—m+l
n—m+l wo >
=$(a)+8,0"S,, d, (a). = went (18).
239. Theorem.
For,
€
eB
88
~~ (s.—) ke (106) and (9):
© m—1 77)
=Boot?.077 As “, (6) and (235)
co
=§,,0"-*.b,-2, Suppose.
1
But —-1=-
2S,,6
@
=6
e”—1
+
e—-l e¢ *-1
©
er Sn On-2)2™ "+(—2)"-*3
Co © }
Seabspalae —3 _ perm +) Se yi Steep sa)
2m—2° ee”, (6)
©
=2h)+28,, Dom Wen (9).
me b=-4, and by, =0.
1 o 2m
= 2n—
~=a = 24890 wa
=]
+4
(14);
).
1
nt ge a, When ia (6) and (228);
) m+L =j/
=o" '-$48n0". Ay, ( ), (9) and (64);
ee} 2m ill
=07'-448,0""). rl J; (14).
2m+1 ill
240. Cor. 1. A.( )
[r+1
2m = Fl
241. The number A..(=5) is called the (2m—1)"
|7+1
number of Bernouilli. In order to deduce the successive
values of these numbers, we have
a ei Ws ne
ie 9 | easel eek ee seme yf ae
| —)- "|2m—-2n+3- of =
~
Ay » (14);
m =f 2n—-1 =
+92. r )
|2m—20-+2 r+i
m =I
=S, eo
2n—2
A. » (240);
2m —i|
and, putting, A... ( ) =Cem-1s
90
1
Es= =~ = 000033068783
67
3
6; =- [10 = — 0000008267 19
10
e, = Tie a 000000020876
691
Cnu=- is = —,000000000528
ie 2
= = 000000000013
63 12 [13 3
3617
=e = — 000000000000.
242. Theorem.
, Bi aime
Ds Uu=h*. fu— 5 eS UY Suomen Rel tr.
co
For, assume D,-\uw=S,,h"-*.dn_2, where dm_o, is some
function of «, and independent of h.
*. U=,h"-"_D;.G_-2, (136) and. (137);
=Snh Sn de" Gyo» (199);
ce m d” a
SS) aie ee (6) and (18);
m+) n
d, . Gy, —n
=d,.0_,+ 5,2" -S,—
i Co 3 (9) °
91
1 Ueda. y, OF G_-= fn
m+1 n
d, . An, =
and 0=5S,, E
™ dt a. =
os ae a (9).
A little consideration respecting the form of this de-
velopment will convince us that d,"w is a factor of d,.a,,_)-
(-1)"*'
|m+1
Assume, therefore, d,.a,,_,=d,"U. 10-5
then d,.a_,=u.(—1).6, and =w. .- 6=—1.
=| m+1 m | m—n+ 1 b, ms
Also teen Cali ING Gin gie wee ) m—n ;
|m-+1 [m—n-+1 [nti
by substitution;
_am ue (=F ar Onan
SO Senet [RT
Wb =) aoe Ge!)
Aa Lee tai piel etemaen Sai
» (6).
( ae i 6, ™
a aby Ae,
and
=-Al ical |
"m =r
oe d,«Gn—\=A,"U. Very (=) 5)
a
m ae
and pe — Oy 1 U : Ax, an
Hence De u=h-?. [.u4S,h"). du. A ay (—)
243.
244.
92
r+il
2m
) alee =
=A. u— — a 2m—1_ gf 2m-lay (—
fru Z +S,,h i bes ay te (ea: (240),
U i=]
=h-*. fu 3 Houcamat weasel, | (241):
Lp es ;
A, 'u=fu- 2 +S .Gem Ge dee
grt a” co
=!
—— +8 Eem-1- m a” m+14const.
n+i1 2
2m—1
CHAPTER VIII.
ON THE DIFFERENTIATION OF EXPONENTIAL AND
CIRCULAR FUNCTIONS.
245. Tueorem. d,.¢*=e".
(106) and (9).
(193), (191), and (210);
DAG. Cor. 1. <d 6? =e.
Qa: Corn. 23 die —e-. 0.1, (202):
943. “Cor. 3) )0,-.e. =e 1 Ok
249. Cor. 4. da%=d. en")
=" log,@ : (log, a)"
— (ghe : (log, a)".
1
250. Theorem. d,.(log.«)= =
For, v=6'%«*
. 1=6'%*.d,(log.v), (197), and (247) ;
=wv.d,.(log, 2).
1
. d,.(log.r)=— .
xv
94
Dae Cok. i. dp peeve 4 Wen).
u
Dba" Con 2 aeiler yng =
log.a
Te .d,.(log.w), (196);
atenl du
~ log.a u
253. Theorem.
ae
log.w= ae 1)" ye =
1
For, d,.(log.v)=—, (250);
av
a" 1
Coan , m being any number;
= 8, (—1)""'(a"=1 1D eee He's ie (a" a, (12);
Fpl Yr ‘
A. —y)=-(-1)' (192), (210),
1 T | m—1
eta )
” Mm
(191) and (207);
e (-1)""? aes —1)". — +const.
1
n
Hence
1 “wn i me aa r
log. 1 = = ; Sn (-1)"7?. ( ou Ee +(=1). l= +const.
and log.v=—. Gi 1"! ae ‘ (a” (ary
m
v2
t=] av
95
254. Cor. 1. If a®~1<1,
then log.w=
ise (12).
Sir
2554 (Con. 2. > If reeks then
es ah
ee ae
rns
log.(1-#)=
1 p
and log, — =log.(1+#)-log, (1-2)
—x
nin m
v af
-S, (= +3n =
=S, {(-yrtays, (5);
a™ nes 1
256. Theorem. log.a=
1 Ces 1
For, log.«= ai fog. —log, sat
S (-1)""! ( a” a
“i eae w+
a) (ois aia. (54-1) | (255);
m “+i
eee S hee en 1) ae 1 bee Er .a"
n m(a"+1)™ me m (41)
is an” —1
m
‘pk m (a"+1)"°
(5).
257. Theorem. p(c*)= “Seo {P+ A)por-).
For, ple)=piite-1}
a di’ -p(t) 2 n—1
Smee; =A, (198)5
n—l
1 a a3
=a) Ae san) 5 ck (- 1) 1 ear Agee es (86) ;
1
Ges 1 t |2—1 ° m—1
=o) 7 se ene ve —1)"- 1 = S,,(2- —ry"- 1 a ae
2 at ee O10) g Lee
Sta ae
© gy") o Gat! t
=S,,. Sa p(t) Ne 1 gr- MS (150) ;
orange alee
se Gisemn
esi Sp(i+A)jo"-", (198).
-) gntm-i
258. Theorem. (¢?—-1)"=S,——— -A*.0"".
|z+m—1
ea) a”)
For, (c’—1)"=5S,, FOREN pall CLE) 3
=) m—1
=S,, vd NOt
|m—1
n am} o F n+m—1
=S,, lf Moses. wv NP Ora.
| m— |z+m—1
ne) pitt = ]
=Sn .Aror+™-!, (146) and (138).
96
7 ae 1
"\n+m—1 1
, (106);
BC) prmecrome GL) 5
(9) ;
iN 1 .O"- 1
2
(146) and (138).
No on-}
, (255), (146), and
1 i) an} m 7
Theorem. pie =e 9 Seca)
oe 1
= ont. (257)
B= lanl faa ao
fo) an} 1
=S,, on-}
m—-1 \2+A
a) a™ 1 Ie 1 07> 1
=Sn ——.§,(=1)"" : gat ea?
oo an! m INES 1, o”- 1
=o pat Sn(-1)"" : a
ve 2 ie Le
Theorem rae =Sn [mi S,(-1)""!
log. €*
= rae
ao aah tlo | 1+A
Set POTN on, (enn):
m—1 14+A-1
ere nS, ; ed al
"|m—1 A
2 ym m Aso")
aD . —1)-!,
S.—.8,(-p.
; xv a 5
261. Cor. Since 1 =1 ie: RUS ig Oe Eee (239) and
(241) ;
2m +2 A®-! 0741
*. SG rie weet ce
m+] NOs 1 Qo?”
Se iae ————=|2m. Coe 5, ane
wv am 2n+1 A? —1 or
: =| at : .
| 2am Sn(-1) n
fos]
av =]
INES"
n+m—1
For, D,?u=(e"4 —1)".u, (201);
262. Theorem. D2 Uu= Se
5 Ses Mer! zm
L? 0” +m—1
Tes : Se Pa eS Ee (258).
263. Theorem.
=S, ———
plog.(1+A){".0"=0, (m2); and
Slog.(1+A){”.0"= | 2.
co a}
For, Sujppmz tlege(1+A)}"-0" "(loge eV, (257);
=a".
. flog, (14A)}".0"=0, (m2n), and LOO +A) 0
[n =i
264. Theorem. Slog. (1+A,)} u=d,w.
co AS
For, log, (1+A,)} w=S,,(-1)"71. =, (255) ;
co (—1)ta
aS) meee (c& — 1)", .(201)5
oo
S (=H © A” onte-!
=n ane
mm "|m+n-1 a BCP):
qn m (-1)"~”
i “m—n+1
Gi? BN eel ay (6) and (is);
pane =) mn Qo”
‘1 (is eu’ (8) and (13);
d,"u
es ae
=d,u “(268).
-jlog.(1+A)}o”, (255);
fs
265. Cor. d,’u= Slog.(i+A,)}"w.
sin 2 tan
266. Theorem. ( } =1, and { =1.
v
«2=0 faa (8)
Lv
taney sing 1
For, = ;
we ® COSe
(= *) (=)
v | x=0 @ TS x=0
Also, for every finite value of wv, tanv>a, and sinw<w.
And these two relations can only be made to agree by
the equations
tana sInw
=1 and =|.
wv 7=0 v 2=0
267. Theorem. d,.sinv=cosa.
sin (w+h)—sinw
For, d,.sma= ¢¥{—————_
h h=0
DY seh
cos { v+ = }.sin=
91 S|] r
|
=cosw, (266).
rr
100
268. Cor. 1. d,.cosv=d,.sin G -«|
= —COSs (= -), (202)
2
=—sinw.
269. Cor. 2. d2*-'.sina=(—1) ‘cosa,
d,°".sinv=(—1)’.sina,
d,?"-' cosw=(-1)".sina,
d,?".cosv=(—1)".cosw.
270. Theorem. d,.tanv=(secx)’.
sinw
For, d,.tanv=d,.
COS&
({cosx)*+(sinw)*
. (cosa)*
=(seca)=
271. Theorem. d,.seca=secx.tana.
For, d,.secv=d,.(cosx)'
=(—1)(cosw)~*.(—sinw)=seca.tane.
1
Lheorem. d,smiqat——j————-
V 1 lee
For, #=sin(sin~' 2).
2472,
°. 1=cos(sin~*#) .dgsin( “a, 4, (202);
= J/1 ait SiMe aie
Sod,.sin =
101
273. Theorem. d,.cos~'#
=
v= si: = >
For, v=cos.(cos”'2).
1=—sin(cos-'wx).d,.cos~'a
=—/1—2".d,.cos ‘x.
—1
. d,.cos~'#= ———.
V 1—2*
1
274. Theorem. d,.tan-'!x= =
1+a@
For, v=tan.(tan~’.2).
7 L=sec.(tan> a)? d..tana' »
=(1+4°).d,.tan7'2.
1
-. @,.tan v= =.
1+
275. Theorem. d,.sec~'x=
I
& VS a > 1
For, #=sec.(sec”' 2).
*, 1=tan(sec—'#).sec(sec—'x).d,.sec™’
=f v’—1.0.d,.sec'a.
1
-, d,. sec” a=! = ——.
v Jf el
av.
CHAPTER IX.
ON THE EXPANSION OF CIRCULAR FUNCTIONS.
276. Turorem.
n — n — nr
P,.(cosa,+4/ -1.sina,)=cosS,a,+ / —1.sinS,v,.
For, (cosa,+4/ —1.sin2,)(cosa,+ +f —1.sina.)
=COS (v7, +2.) + nie sin (av, +22).
And the introduction on the first side of the equation
of a new factor of the form cosa#,++/ —1.sinw, will increase
the arc on the second side by the quantity #, Hence the
truth of the theorem is manifest.
277. *Cor ty Patrx,—z@,) then
(cosv+4/ —1.sinx)"=cosna++/ —1.sinnx, being any posi-
tive integer.
Again, (cos#+ Ny A sinx)(cosw—/ —1 .slag)= 0.
“. (cosv+ BY Si : sinv)-!=cosa— —1.sinw
=cos(—a)+ rf -1.sin (-2),
and (cosa+r/ —1. sinv)~"= } cos (—a’) + ne sin (—«){”
=cos(—na’)+4/ —1.sin(—n2).
103
Hence, (cosa+/ —1.sinav)~"=cos(+ma)+/ —1.sin(+ma),
m being any positive integer 5
m : m
=cosfn.(+—]at+r/—-1.sin}n| +—)at,
n : n
n being any positive integer;
m ry sg m :
= {cos (+2) +/—1.sin(+7.0)|
n n
ee m — , m
*. (cosa+4/ —1.snv) "=cos| +—.a@ +/ —1.sin ~£—.v}.
n n
Consequently, whatever rational value 7 has,
(cosw+ ait sin v)"=cosnv+ / -1.sinne.
278. Cor. 2. 2cosnwx=(cosa+V/ —1.sinz)"
+(cosm£r/ —1 slp),
and 24/ —1.sinna=(cosa+/ —1.sina)"—(cosa#£/ —1.sina)*”.
n—2m+2, 2m—2
n lo=)
279. Theorem. cosS,«,=S,(-1)"—'.C,,;(cos@,.sina,),
n—2m+1,2m—1
and sin S.2,=S,. (-1)”"'.C,,,(cosa,.sina,).
n —s n
For, cosS,2,+ rf —1.sin S,2,.
n =
=P, (cosa,+ WA —l.sina,), (276);
co n—m+l1,m—1
=§S,, C,,; 5 (cosa, JOST: sinw,)}, (67) and (13);
m—l ers m—1
= 255 (-1)? Cz; (cosa, sing), (55);
mire 2m—2 F
48..(< Dre !.€-.(cos#, sina.)
n—2m-+1, 2m—1
hf SY. S, (-1)"-'.C,,,(cosa,.sinw,), (14) and (6).
104
Whence, by equating possible and impossible parts, we
obtain the above theorems.
ag 2m—1,
n m 1
280. Theorem. tan§$,#,= Sa(=0"=!-C, (tana)
2n—2,n
Suet y"-*.C, (tana,)
n n
For, cosS,2,4+Y —1.sinS,2,
=P,(cosz,).P,(i+-V —1.tana,), (276);
m—1 m—1,n
_p. (cosw,). Ss. (-1)? .C,(tanaz,), (68), (55) and (13);
2m—2, n 2m—1, 7
=P, (cosa,) {S,(— 1)""1.C,(tana,) +7 — ne 1)"-1.C, (tana, )s
(14) and (6).
Hence, equating possible and impossible parts,
QIm—2,n
eee (cosz,). ve 1)"-1.C,(tana,), and
2m—1,n
BAC ap (cosa, )S,.(—1)"" 1_C, (tan 2,);
and, by dividing the second of these equations by the first, we
obtain the theorem sought.
281. Theorem.
n—m
cos na=kn. Sy (<1)? 72 cs (2cos x)"-°"*?; nm being any
integer. era
For, 2 cos nv=(cos v+ —/ A .sin wv)"
+(cos w—af i SHLD) > 4208) 9
= Cos nx (cos v7+4/ —1.sin @)"
oe —2 2 Ss ee 3a
=a (cos x—*/ —1.sin ay
n t"
¢ being any quantity greater than unity, (4) and (6).
105
1 cos @+%/ —1.sin & cos v—a/ —1.sin a
= Og. | b= a +log, aa a aaa
(255) ;
=log, }1-t~-'.(2 cos w—t7')}
2a(2 cosia—t,_)*
=— S,, arn A 5 (255) 3
n
bad ee a) m— 1
= sae si a . (2cos #)"-™*?, (86) and (13) 5
| —m
=-S,—.S,,(-1)"7! ———. (2 cos w)""*"**, (6) and (18).
Ga |m—1
Hence, equating coefficients of ¢~’,
nm—mMm
n
=—Sn(-1)"". ea (2 COS a) we aa ts
:
—2COSnx
x | 2 —m
nite iy eee ae (2.cosm)ta <7, | (13 Y.
—_
282. Cor. If mis even, every term will vanish after the
(4n+1)™, and
| 1 ee
$atl gnt+m 2
cosnv=4n.(-1)'".8, (-1)" "| eee. (2 Cos)?" ~*,
|$n—m+1
(8) and (6) ;
106
|dn+m—2
; |2m—2 [an—m-+l
be: Weta
= sis n+m—2.4n. =
m—2 es
“lem—3° |$n+1. [an 1 (40);
m—2, 1 ?
ah 2 m—2
GO” 'Pn+r\(hn—r)|
. cosnav=(-1)'".” Oh ae 1D ie a ae (09
Im—
If m is odd, every term will vanish after the }(m+1)",
and
$(n+41) IE (n-1)+m—1
cos 2v=4n.(-1)*”".S,,(-1 mo) so, (2 cos wv)",
|$(m+1)—m
(8) and (6) ;
107
Ts (iy> ea 1i(n- =
Ges ME EE (1) ra a on Dts
over [pe it) a 2 (n—1)—m 2m—2
~ [S(Q+1)—m ~ am—1. |S (u+1)—m aca T
4(m—1)+m-1.|4(@-1) | $(m41)-|F@-1)
m—) _._m—1,1 m—1
2m—1 |gm—1
m—1 m—1
P, 3 (Gn+r-)Gn-rt+ si PG) Gees
2m-1 |2m—1
P, {n’-(2r-1)*?
| 2m—1.g?m-2
m—
# (n+) P. \n?—(29 as
. cosna=(-1"? .n.S,,(-1)"71. (cos @)?”=?,
2m—
283. T'heorem.
zn [me
(sin @)y=(=1)".2-**" Sate) _—=1_ cos (n—2m+2)u
|m =F
|
ae a — AOL
roy VL
n
4(n+1) —
(sin wy"=(-1)8"? 2-741. 5, (- 1)". sin (n-2m+2)a;
according as ” is even or odd.
For, 2 AY) a . sin v=(cos r+ pn . sin 2)
—(cos 7+ Vf —1.sinw)7'.
108
(1) If mis even, then
n
$n
g"(—1)'”. (sin oy a ea
{(e0s a+4/ —1.sin vy"
nu
1
—1)?"_ 22 :
ane aes} +( 1) 3 A? (90) ;
nN
zn
=S,,(-1)""! —_ 2¢c0s (n—2m+2)v+(—1)*". in (278).
[m=3 [an
*, (sin 2)"
(2 I”
zn
(9) 82 SEA On ee a COS (m—-2m+2)0+2—", Sm
2
.
=
(2) If mis odd, then
n
+ (n+1)
g” .(—1)'”.(sin eat Tae |(oos Da-a/ —1. Sin)
1
=a 5) (90);
(cos 7++/ —1.sin 2)
2(n+1) &
=S,,(-1)"") 2a. 24/-1.sin (n—2m+2)a, (278).
—1
|
- es s\n. 4(n—1) he m—1 m—1 -c ; a
. (sin #)*=(—1) .2 ‘Sn(-1) Seam (m—2m +2) x.
m—tI
109
284. T'heorem.
ie | 92
zn | is
+1 cos (m-—2m+2)H+4+27 a ~ , or
2
1
(cos #)"=2-"*!-S,,
=
n
cos (w—2m+2)a; according as 7 is
even or odd.
For, 2 cos. #=(cos v+ / -1.sin xv)+(cosv+r/ —1.sin v)~!
(1) Ifmis even, then
n
2" (cos x)"= “5, SH (co wEA/ Si .si ay
ay |”
aE iis _ (89) ;
(cos w+ Af aissina)? "=? |5n- ;
gs :
=S,—M—@—. 2 cos (2—-2M42) e+ am Q78) :
met ( ) Ee (278)
8 | : |
(cos #)"=2-"11. 8, - = .cos (m- 2m +2)V+ 27" —
|m—1 lo 2
(2) If mis odd, then
2 (n+1) |
gn (cos oy Buea | (co et Ay ee =~] , sin a= 2m +2
Mm
Hs (cos C+ TS sin aya 4 ? (89) 5
(278) ;
2cos (n—-2mM+2)a,
in
4(n+]l) Lo
m—1
cos (2-2 +2) w.
. (cos n)"=2 =-ny S,,
fn
110
v 2m—1
285. Theorem. sin «= c (—1)"-". :
2m—1
co an 1
For, sinv=S,, 57 eae ‘sin a, (213) ;
"ae
2m — 2 i) yrm- i
2m—2 _s :
dC sin ree
i=)
ye zm—-1 .*
=8, ome te Ne ema Sin @,
m—2 In —
oy C-) an il
(14);
co
=S,, . (-1)""". (sin 0) ae . (-1)""! . (cos x) =
2m—2 ‘ |2m—1 Rae
on) 32m —1
=d,-Sn(-1)"7 yeas
prm—2
at m—1
23% 1) |gm—2
287. Cor. 2.
@VA=S, (-1)"- i
(14) and (6);
=cos v++/ —1.sin a,
Se 5 / Sie ye
(So Sok (ea) [2m
and ¢~*¥-l=cos (nye n/a .sin (
=COs a—/ —1.sin Ds
e7V-14 ¢-*V-1=2 cos a,
and ¢*V-1_¢-*V¥-1=2a/ —1.sin wv.
288. Cox. 3. ¢®-Vie. VA=(-1)"1.4/@1,
and (mer Vet (1.)°=1,
| i
m—1’
—x)
(106),
111
289. Theorem.
oS n 1 m—1 — jl
at _p\n—! . 2n-1 a oasis | |
tan x=§, ( 1) + ‘Sn Sn —omal he, ( |
For, put s=—2’, then
tan v=sin v.(cos v)~!
ea) gral co gm) -1
=v. 285) and (286) ;
ApremlGapeey 2 ae iGey
a) gn) eo
= S., aes Oe oo"! Gas
1
where @,_-)= [am at (228) ;
— f n 1 ; m—1 —A, A
S Out tn Merry ag ct oe: (=) ; (235) ;
© n i! m—1 ill
rd —1)"-), @"-}. ee 5A (xd Fr
S. (1.08. aaa Ae (a
cc) n—1 —ji
290. Theorem. sec 7=§,(-1)"7).0"-?.A,, (a)
For, sec 2=(cos 2)~'
z (S, oe ss (286) ;
eo
=S,,2""'q@"-1.a7', where a,_,= [en-2 5 W289 g
co : n—-l —j|
=§,(=1)*) a7? AL, (=) FE:
291. Theorem.
roa) n 1 m—1 1
a2: (aye ame (i es son)
cot #=S,(—1)" 1 Sn |2n—2m Ass ars
For, cot v=cos v.(sin 2) ~!
co gnol o gmt -1
= .{a2.5, —— 285) and (286) ;
Since: (#5 =<), (285) and (286)
oe) gn ee)
1
|gm—1°
where @,,_1= (228) ;
oO a n i] m-1 ji
= S.C) ha Sn (Ve ( ) C
|2n—2m
© na—l =
292. Theorem. cosec r=S,(-1)"-!.a"-3.A,, = é
|2r+1
For, cosec v=(sin x) ~!
o gn l -1
= (7-8, — 5 (285) ;
= 1
=—«¢-'.§,3"-!.@"-1:a7-1, where a,_,= [ana , (228);
n—
rae) n—1 a
pe Sie?t! aa Va (235)';
293. Theorem. sin vaw.B,fi- (2) 1.
For, the roots of the equation
O=sin v
r=]
Tr—Olye
are @=0, and) g=——r7, (
o
-. sin v=aax.P,}(a—r)(w+rm)}, where a is independent of 2 ;
=a2.P,(-rx).P fi ( z )
TC
o
=ax.P,(-r'n’). {1+terms in 2°%.
But sinv=#+terms in v*, (285);
i-=)
- @.P,(- 9) =1,
: 2 a \?
and sin v=27.P,21- (=) :
vor
= 2H 2
294. Theorem. cos e=P,f1- (=) .
For, the roots of the equation
O=cos w
are w=£(2r-1)7, (‘=)).
ee Qr-1 : ee
*. cos =a. P, {a (—. ) \, where @ is independent of a ;
~
«Pf (=: a) |B {i- (—=-)}
aP,|- (== vn) }1+terms in at.
BE
Hi
114
But cos v=1+terms in #*, (286);
or. eae
© Qe \*
and cos o=P,|i- ( } .
2r—-1.a
a2) 2m
295. Theorem. log, sin v=log, -§,, (=) ;
co 2
For, sinw=2.P, } — (=) \, (293) 5
no
H co xv 2
. log, sin a=log, #+§,, log, ‘ = (=)
re) 9 1 a i)
296. Theorem. log, cos v=—-§,, (=) .— .§,(22—-1)-*".
7 m
2 20° \"
For, cos =P, {1- (—— ) \, (294) ;
2n-1.7
2 9a \*
-. log, cos v=§,, log. |1— (
2n—1.7
=-S,S, ( a \ es (255) and (6);
2n—-1.7
=-8, (*)".—.8,@n—1)-, (17) and (6)
115
297. Theorem.
co Lv 2m 12
log. tan alos: v+S,, ( ) ee ase (- 1 ies na",
T “m
For, log, tan v=log, sin a—log, cos #
e) Qn 2m 1
HS (=) Oe
Tv m
ay ee ea
=log, v foe (=) °ss
T m
298. Theorem.
n
tan7! #=8,q(-1)""1.>
(274) ;
=
For, d,.tan =
.S
=S (-1)""!.
.. tan“! v= ae (-1)""'.
——+(-1)" foes
999. Cor. lf «<1; .tanq
fo]
S,(- i Yee ie.
v&
pers =a nr.
(ay
io=)
1 G=S,(-1)"-'.
-) Qu 2m 12 ;
-Sn =) Sime Ss: (2 n) Pa
7 m
fon)
.9,(2n-1)7"
(14).
aie: ane caer
27
(2);
9?
a”
, (191) & (210).
s2m — 1
u
Im—1
116
300. Theorem.
coat re yr, 2 — -S,(-1)""!
WT
tan 4a@—tan Fi
T
1+tan be tan
Tv 7
I Sin (40- =
A A.
1
=4,tan~'— —tan~1——
5 239
1
fo=)
boa SN) eat (agai)
1
We ky MAW nk ee ee
Sit ) (250)">" =
But —~ =4x(,2)!"-!=,8% (04)
Bem -
ba - (,04)"- i :
2,88, (HI Sy (1).
]
BN")
(299).
(2m-1)’
]
Co aaKCEDS
117
2m—1
, vai
301. Theorem. «+1 Spe (vw-e ™ ee 54)!
For, put 0=z"+1; then 2”=—1
= _@ Narva, (288).
2m—1
Tay
. v=e ” ;
2m—1 —
and the ” different- values of « * °”*— are the roots of the
equation
O=Ne-tals
ae
Hence the » different values of (a—-e ”
factors of (#"+1).
mice ~) are the simple
302) Con. I:
2 2m— 2m—1
a"+1=P,, §a—(cos r+4/ 21.sin-——— ‘7) ty « (287):
n
303. Cor.2. If mis even, then
1
zn 2m—1 2n—2m+1
aN —1
mr i=P, (ate = ~Y) (ate n : a2 ),
by inverting the order of the latter factors, (31) ;
zn 2m—1 — _2m—1 ny : a ae
+e ” *)+1}, since ’™V-1=1, (288);
in 2m—1
=P, (2-22. USS raene .w+1), (287).
If 2 is odd, then
+ (n—1) 2m—1 $(n—1) 2n—I2m-+1 —
av "+1=P,,(¢— Fae ee ra Tay (ace). P,,(e- € % le
(32), (288), and by inverting the order of the latter factors, (31);
2 (n—1) 2m—1
= (v+1) ' iP Sa —a’ (e Fs amas
since’. e?*V=1= 15: (288) :
’
4(n—1)
=(@41): P,, (v’-24. does
118
2m—2
ne Va
304. Theorem. «x Sos) aera ).
For, put 0=#"-1; then a”=1
Sere ss (288):
2m—2 Z
xe v=€E Le 5
2n—2
. ——_ .7V-]
and the m2 different values of ¢ ” are the roots of
the equation
O=a2”"-1.
2m—2
: —— or V=1 ;
Hence the n different values of (v-e ” ‘"*~') are the simple
factors of (#”—1).
305.7 (Cor. a.
z 2m—2 . 2m-—2
a"—-1=P,, Sa—(cos w+ —1.sin .m)t.
306. Cor. 2. If m is even, then
zn 2m—2 Na — = ; ‘ 5
w"—-1=P,,)(a—-e ” )) Ceara )i, by inverting the
order of the latter factors, (31);
tn—1 Qn 2n— 2m rn =y i
=(v-1).P,,§(w-e” Sate )} (w@+1), (82), & (288);
¢n—1 2m ae 2m. ER f : oa
=(0°=1).Byfaraler pe 4}, since V1
zn—1 9)
5 ~m
=(#?—1). P,, (2-22. cos. -w+1), (287).
If is odd, then
+ (n—1) Qmn7 — 2n—2m =
eS UA S| : .
v"=1 =(«-1).P,, ; (w-e n '\(a-e n i ) t 5 by inverting
the order of the latter factors, (31);
+ (n—1) ae SE
=(v-1).P,,ja’-av(e" “+e ‘\+1hs since eV 15
4+ (n—1)
=(v-1). P), (a? —20. cos —
119
n
307. Theorem. (i+e. cose)"=14 4. aa ($e)
(my
- n
+2COSM#.S, ie (Leymt2r-2e°
> = Nepen o
ae is |
For, a at ee aed (92) and (9);
n n
o
=1+8, a (ecosa)?"—" ieee omar (14) and (5);
|2m—1 2m
| 2m—1
eo ™m
=1+58,, m= : en) —_ .§, cos (27-1)
es a
| 2 : 2m 2m
— m
am 2 —r
+ —e "(= ae cos2rav+ a at (284) and (8);
| |2a—1
S, L.2.(he)"! S$, cos (27-1)
24g 8 SS conor).
n 2m
o — m
2m 1 py\em m—r mas
+3nToo .2.($e) ge pacer COS2TA
n
See “(ae (Le)
| 2 .|2m+27r—-38
S 2cos(2m—1)x. ea Sea le 2m +2r—3
ss n .|2m+2r—2
+2cos2mv.S, == Gey re
"[2m+2r-2. (aaa
(19), (17) and interchanging m and +;
120
|
ts = n -|m+2r—-2
+5 ,2COSmx (yn ter :
Sn = Enea pet .(4e) » (14);
I”
SES, (i --(4e)"
i? n
+2cosm”. Sa Ce eel 1d pyn+2r—2
S: m+r—1.|r—1 Ge » 5)
308. Theorem. If y=x+a.siny, where x independent
of x, then shall
Y=r
| 2m— a
+28,(-yr. G9 — §
m
—S(-1) —
em
ara pe
[mar VE
© (La 2mm oa
eo) ae ex iS) (Ea 1 a (2r)- Le ogee
*.sin (2r—1)s
© an 1
For, y=2+Sn ‘emai .d2"-*,(sing)?"-!
m—
o arm ; : ;
+S, iam .d2"-".(sinx)*", (221) and (14);
© 2m —1 en | 2m—-1
md .g2m-2 ma Tak
+Sn REE Powe fon 4)" Si(- ») [mar a .sin (27— ns}
S Tipe a i m | 2m
a" m—-r _m—r Oise
+ Om |2m z tee 4)" or (- 1) lear
2m
4 ™m
im a | (283) and (8);
121
2m — 1
v
"|a@m—1.(—4)"7} 1.(-4)""?
ous
a”, 2 m
S. eC eG arisne ta
© (ga)""* — 1 m
=242:,,(—1)"7- |[2m—1
Gay" 2
EES
309. Theorem.
+2Sn(=1)"7).
cos y=cosz—w.(sinz)”
ue athey
Tex 1
‘ay
(s i alae < (-1 ee 1
|2m+1 Z
Ue 1)”.
+8,(-1)".2
oe
DB
=)
For, cosy=cosz+§,,
i=)
=cosz—2.(sinz)?—
fn} ae +1
ae
Q
S,(-1" a
eG 1). Gr
Bye
; d= ail (sin Ey ier
22m
* [=
i am : (sin sem nae
2m—1
(- 5) ier 1 .(2r-
1) Sera:
sin (27-1)z
a
——— (—1)”.(2r)*""'.sin2rg,
-r
(269) and (202);
| 2m—1
=D peo.
sin(2r—-1)z
oa
(9) 2m
(27)"— snare
In the same case,
| 2m+1
m—n+l1
| m —n+1
2m+2
m—n+1
“|m=n+1
.d,"—'.S(sins)"d,.coszt, (221):
.(2n—1)*"~'. cos(2n—1)2
(2n)".cos2nz, (6).
(268), (196) and (6);
dy 2m—1 .(sin syne
(9) and (14);
=cosz—a.(sinz)”
2m+1
arm m+1
2m+1 |
2. ed RD ee SIN 290 — at
[2m arsed [ecner rae
aes |2m+2
= ee. 1)"-"4) ; mt. COS2NZ
Sta - | 4yntl 5.(-1) “|m—n+1
|2m+2
+= acral, (283) and (8);
m+1
=cosz—2.(sinz)*
am <i |2m-+1
Sei fz ae S.,(- by Nh m—n+l1 <(=1)" Una):
m—n+1
cos(2n—1)%
eee a a |2gm+2
“Sen (coe ee Oe
m—n+1
cos2nx, (196), (191), (269), (202), and (193);
=cosz—a.(sinz)*
be ba’ |2m-+1
ae
1 n—1 m—n-+1
OO nent
: .(2n—1)*"—*.cos(2n—-1)%
+S, (-1)".
e 1) eae m+1 |2m-+2
m—n+l
Se -1)”. Jemqa on ae Tie (6).
CHAPTER X.
ON THE INTEGRATION OF CERTAIN DEFINITE FUNCTIONS
1 —_—__
310. Turorem. lbp ae =log. (w+ rf w1).
e Jf a £1
For, put #’£1=’;
then v=ud,u,
and w+w=u(i+d,w).
1 i+d,u
uu v+u
1+d,u
and ie =f =i
2 C+
=log.(w+u), (251);
© | ~——— slog, (94/41
or lox og (w+ ve );
a
311. Theorem. Leite =loo, ——_———..
8 a n/ 1 La?
(510) and (202);
124
1 d
312. Theorem. a : =log..tan =.
smn &@
1 siInw
sine 1 —(cos x)”
ie 1 1
Sy NE ee th ee ee
l1-—cosx 1+cos#
fai d,.(—cosx) “ d,.(cos 2)
2) 1-—cosx 1+cosx
| (268);
1
; J Fay =F Eoge(1 cose) —loge (1 +6082) §, (251);
1—cos@#@
=log. \/ ——_
1+cos@
v@
=log,.tan rf
1 b
313. Theorem. f[ =log,.cot. a — = :
2 COS@
F 1 cosa
or, ——=-———,
cosa 1—(sinz)”
1 1
=F .cos2{——____ +} ——__
l+sing 1—sinw
,fde-sinv d,.(—sina)
2 \14sine 1—sing
\, (267);
1
: =1 Slog. (1+sinx)—log.(1—-sinz)? 251);
[ego Moge(1+sina)—loge(1-sinz)§, (251)
1+sinw
=log, ——
1—sinw
314. Theorem.
— anf 1-2
: = m1y/) 2 +(n— 1) ey id
a” Le 1
and ==
@
1 —v if v 17 . / 1 —v ;
a” r yr 1—2(m—1) m—l Malo Cs—1
af Ge | ee ee ne joe
fine n—2 n—2.(m—1) 1) m—2.(s—1)
r oe ane ;
+P. m—2.(s—1) ae (51);
av
[m-1 m1 y— er
as = 4 alse —2 = ar iso 6).
= _/1—a". 4 ria Bae em ea ( )
m,—2 T,—2
315. Cor. If 7 is an even positive integer, then
n—1 1
n zn
i Tit CN eS Si BE gn?mt1 4 ee sinter, (40) and
: nr
aeal — 7o [2 g
a m,—2 EN; 2
126
and if 7 is an odd positive integer, then
a # (nt) L— :
{>=-=- 1-2 .S ——— . gon),
2/1 a | %
316. Theorem.
n—2 n—2
oo ee 1 1
Bese / 2 m—1,—2 %,—2
1-2 oa ssl +. [ : ae ;
1—a77 mA
m,—2 7,—2
a?
—a"—ha/ I-ah+(n-1). fF
ve 1-a#@
For,
a”
UERG. | ee
/ V 1 — 2
a a me, nN gp (n= 2m—1—-1) ae ? n—2—2(s—1)
>V1i-« "| n-2(m-1)-1 *(m-1-2(s—1)
EG 6
n—2 |~—2
Ear m— m—l,—2 1 r*5—2 [ 1 >
On AEP : [Sea at Jn 4 7 Ee aS me = 4 (6).
mM,—2 r,—2
127
317. Cor. If m is an even positive integer, then
hee ‘ 5 ae ‘
ree 3S "| |n—1 a" am+1?
m,—2
and if is an odd positive integer, then
ae / ee 1 an b v
m,—2 £(n—1),2
ae and (311).
318. Theorem.
Sg (ae n—1
f.sinw)"= —cosa.S,, ==. (sinw)"-2" +14 22. f.(sina)"-*".
Ee
m,—2 r,—2
For, (sinv)"=—(sinz)""!.d,.cosa, (268).
. f,(sinz)’=—(sinz)"~!.cosa+ f,(m—-1) (sinw)"-?.(cosx)*, (222)
and (267);
= —(sina)""!.cosw+(n—1).f, }(sinx)"-*—(sinx)"}, (196).
. n. f,(sina)"=—(sinw)"~!.cosa+(n—1). f,(sinw)"~*, and
(sinz)”"!
f,(sine)"=—
. [(sinay'=8 {- Sed a cose Pe
n—-2M+2 —2(s—1)
n—1 ;
.cosv+ ——.f,(sina)"~?.
n
P| «fe (uiaee-*, (51);
nm—2(s—1)
i eal =
=—cose. S., aE ‘ (satay ha? eee or ae . {,(sine)"- nese (6).
m,—2 T,—2
128
319. Cor. If m is an even positive integer, then
nm—1 1
an
[.(sine)"=—cosa.§,, =. (sinv)"- "41+ in w, (40) and (197);
| 2
m,—2 tn,2
and if is an odd positive integer, then
53 ene!
f,(sin oe) = —Ccos2. id m—1,—2 : (sin i ee
i
m,—2
320. Theorem.
jf, (sinv)-"=—cosz. “nat Gna ~ een aE
m,—2 r—2
For, m. /,(sina)"=—(sinz)"~'.cosw+(n-1).f(sinx)"-*, | (318).
. —n. f,(sinw)~"=—(sinz)-"-!, cosa—(n+1) f,(sinw)-"-*, and
n. f,(sina)-"=(sinav)- "+. cosa+(m+1) . fp (sina) <2,
. (n-2). f,(sina)--?) = (sinw)~"-) .cosa+ (m1). f,(sinw)-", and
COS @ n—2
= —(n—2)
ae ee i :
(2—1)(sina)"~! ee J-(sin2)
f[.(sina)-"=—
+8. cosa” 1 IP, eet
~ m—2(m—1)—1" (sinx)"-2"-)-1 n—1—2(s—1)
+P Nf esinay-™, (51);
m—1—2(s—1)
, [2-2 : |m—2
2 io ae es Ss 1,—2 } s] Ve —(n—2r) 6 7
|2—1 (sing)?-2"+3 ri |2—1 fe(sin it) 5 (6)
m,—2 r,—-2
129
321. Cor. If 7 is an even positive integer, then
n—-2
1
f,(sina)-"=—cosa. S.=.
m,—2
and if m is an odd positive integer, then
n—-
boy eee
ie m—1 ,—2
[.(sina)-"=—cose. SF “1 Gina
mM, —2
1
+e log,.tan. “|, (40) and (312).
~
+ (n—1),2
322. Theorem.
ee [nex
f,(cosa)" =sinw. a === (cos Cy atau em
i
m,—2 T,—2
se(eOsa yy.
For, (cos#)"=(cosa)"—'.d,.sina, (267);
*. [,(cos#)"=(cosx)"~!.sina+f,(m—-1).(cosx)"~*.(sine)’, (222)
ae (268) ;
=(cos#)"~'.sina+(n—-1). f,§(cosw)"~*—(cosa)"{, (196);
-. m. f,(cosx)"=(cosx)"~!.sinv+(m—1). f,(cosw)"~*, and
n—2
i) n—1
oo .sin 2+ —— . [, (cos x)
f-(cosa)"=
. (cosx)*-2"— 9-1 sing ? pote 1)- +
n—2(m—1) * | n-2(s—1)
Hpi et iS (cos x) se (51);
eae fea
}
=sinw.S,, 7. (cosv)"?"*14+ ==. f(cosw)"-*", (6):
[z
m,—2 1,—2
R
130
393. Cor. If 7 is an even positive integer, then
n—1 [a
$n,2
zn
[-(cosa)"=sina Sy mo (cosa) "14-22 .xv, (40) and (197);
kes. E
m,—2 $n, 2
and if 2 is an odd positive integer, then
(n+l)
f,(cosa)"=sin®. Gea casa)
ee
m,—2
324. Theorem.
|2—-2 |m—2
é 2
1
cosx)~=sine. 9, 22 .§ ——____ + = {——-
Je( ) ora (cosm)*52* ‘ |2—1 (Cosme
™m,—2 r,—2
For, 7. f,(cosx)'=(cosx)""'.sinv+(n—1).f,(cosa)"*, (322).
2 —n. [.(cosa)-"=(cosa)~"*" .sina—(n+1). f,(cosa)“"*,
and n. [,(cosw)~"=—(cosw)-"*” sina+(n+1). f,(cosr)-"*".
-. (n—2). [(cosa)—°-? = —(cosx)~"-".sinv+(n—1). f,(cosx)~",
(cosn) >" n—2
ee Si pe COs e) ae
ats ee f, (cos)
and f,(cos#)~"=
sin& 1 m1 (7—2(s—1)—2
Pp, pe
* n2—2(m—1)—1 | (cosa)*-? "91 n—2(s—1)-1
ee pe
n—2.(s—1)—1} ~* (cosa)
> (51);
n—1 (cosw)"~2"*! i |w-1 7 (cosa@)"~* i
131
325. Cor. If m is an even positive integer,
. |n-2
in
m—1,—2 I
wy
f,(cosx)~"=
ize 0 (coswy2"41?
m,—2
and if m is an odd positive integer, then
2 (n—1) |n—2 1
.(cosa)~ "=sing. a ea set (cosa)"-2"*!
m,—2
1
then
rage 5 T v
i ~ 1),2 log, cot (= BS a)? (40) and (313).
$(n—1),2
326. Theorem. f,(sinw)"
n
} & sin (7#—2m+2) x
={-] aaa 1 m= Ae) m—l !
) Sn(-1) |m—1 n—2m+2
——__
2
nN
=(=1) 1) 9-1!" (yn, Ke ‘
m [m= nna
according as 2 is an even or odd positive integer.
1
in
cos(n—2m+2) x
[7
o-n punta
Be
-@ Or
For, (sinz)"=(—1)'".2-"* 1.8, (-1)""!.———.. cos ( -2 4.2) &
ata me
3 (n+l)
=(-1)?"") 2-741. S).(-1)"71 sin (7-2m +2) a,
|m—1
according as ” is even or odd, (283).
Hence, by (196), (191), (268), (202), (197), (267) and
(6), the truth of the above theorems is manifest.
132
327. Theorem.
n
Aes 075 fa sin(n—2m+2) a
f.(cosx)"=2-""".S,, ——
[m1 n—-2M+2 ay wae
n 5
donnie i: sin(m—2m+2) a
= . more ee SS ee
|m—1 nm—2M+2
according as ” is even or odd.
” nN
= .cos(2—2m+2)+2-". Soe, or
For, (cosx)?=277". S, Sr
Wk ie
Cae
according as ” is even or odd, (284).
—Q-ntl
.cos(2—2m+2) x,
Hence, by the same articles, the truth of the theorems
is manifest.
328. Theorem. f,(a*.u)
=a*.Sq(—1)""). (log.a)-".d,"-"0-+(—1)". (log.a)-*. [\(a®. dw).
For, f,w.a®
=S,,(-1)""1.d"u. fra?+(-1). (Cu). frat, (222);
S10 ae 2” u+(-1)’. Neo du (249);
=a". 8, (—1)""?. (log.a)-".d."-!w+ (—1)*. (log.a)-*. f(a". da),
(6) and (196).
atl
329. Cor. f,(a*.«”)=a".S,,(-1)""*.|” .a"-"*. (log.a)-";
m—1
nm being a positive integer.
133
a ] m
330. Theorem. [ Elon: “18,
2 U .|m
2 (I
Horees— leo Scie w”, (107) and (9).
Tor) ] m
(logea) (ont wand
a" (w.log.a)”
f= =log. ge S. (250), (191), (196) and
(210).
331. Theorem. [,(a*.w)
~a?.S,,(—1)"". (log. a)". [™u+(—1)". Cog, a)”. (, (a7. [72).
For, f.a’.u
=§S,,(-1)""!.d,"-}.(a*). "w+ (-1)". f.5.d,".(a’) frur, (222);
=S,,(-1)""2. (log.a)"~}.a”. [w+(—-1)".f,§ dog. a)”.a*. fw, (249);
=a*.§,,(—1)"~'. (log. a)"-!. ["u+(—-1)". og. a)’. f,(a".. fi"), (6)
and (196).
n—l yotm
332. Conf =a8,(-1)"' (log. a)".(- Naren
™
sad 1 UR a) ar ee) x ue LO) 3
--a".S, - (log. ay"~" ene et a —, (6) and (196);
ae ms eoetm (lo a)” © (wv. log, a)”
Su! Sn (log. a) , [na aig (ego [m1 flog. L4+Sin ate I,
m
(330).
134
1 on) ] 7
333. Theorem. - —_ =log,’ 5a”
,log, « mM. |m
For, put log, v=u;
1 1
log.a Y,U
then w=e", and A:
&
dx
u
m
= U
=log, u+Smn m.|m” (339);
(log, x)”
—m.|m *
8
m.|m
=log? #+Sn
334. Theorem.
flog, (1+e.cos #)=—a.log. {2e-(e*-»/e*-1)}
2 sin m:
+2.8,,(-1)"7!.(e7!=- fe-?—1)". : Ei
m
2
For, put ae then k=e7'—»/e-*—1, and
~
log, (1+e.cos v)=log, (1+ age x)
=—log, (1+k’)+log, (1+2k.cos #+k’)
=—log, (1+k*)+log, §(1+h.6°V=)(1+h.e-*4)?, (287)
135
=—log. (1+) +log, (1+h.e"¥4) +log. (1+k.e-"Y)
m
= k ae ite
=—log, (1+4°)+8,,(-1)”""'. a (CN fen Nye (255) and (5);
m
ee k
=—log, (1 +16) +2-S_(—1)""-—.cos mx, (287) and (6).
m f
o
“, flog, (1+e.cos x)=—wx.log, (1 +H’) +2.Sn(—1)""!.— sin Me,
(267).
=—wlog, fae(e—n/e?—1)}
sIn mx
$250.41) (er aera):
m
CHAPTER XI.
ON GENERATING FUNCTIONS.
335. Tue symbol §,,@m will be used to denote the sum
of the series formed by giving to m every integral value from
m to r both inclusive; zero being also taken as a value if is
either zero or negative.
336. Definition. If p(t)=8,u,t", then @(¢) is called the
generating function of u,, and is denoted by G;.%,.
337. Theorem. G,(u,+0.2)=G;-Uct Gi. V,-
—a, 0
For, G,(u,+0,)=S,(UetU2)
—aw, & —0,0
HS, Ut +S.%.t, (5);
=G;.u*+G;.0,-
338. Theorem. G,(au,)=a.G;,.u,; @ being independent of
# and 2.
,
For, G,(au)=S,au,.0
<
137
339. Theorem. If G;.w,=G,.v,, then shall u,=v,.
—2,0
For, §,u,¢7=G,.u,
=G;.v,, by hypothesis ;
—;, 6
=S,v,-t.
". U,=V,, by equating coefficients of ¢”.
340. Theorem. t*".G,.u,=G,.U,
lay (3a!
—, 0
a= == 5
For, &”.G,.u,=t".S,u,-
—2, 0
=S,u,.t", (6);
—,
= S. Wren < te (9) 5
(Fria
341. Theorem. (t-'-1)".G,.u,=G,.A,".U;.
For, (¢7'-1).G,.u,=Gy.Up4;—Gy-Uy, (340) 5
Net (33):
Hence, (¢-!—1)*. G,.u,=(¢"'-1). G;. A,U,
=G;.A?7U,3
and, similarly, (¢~'—1)". G;.u,=G;.A,"Uz-
S42. Con. 06 = Dt Gite Ge As ae
S
138
ape An
343. Theorem. ( Se p"-
n+) a ; n+1
‘m—
For, ( Se — : Gy Soe! peo? NG
n+l
=n ta—a . Ges Up teas
n+l
sh
=G;.SpOn—1-Us+m—1>
Sin Om Ve
n+)
G;. U, p= Gy. Sn Amn— 1+ Uz+m-1°
Uz» (6) 5
(340) ;
(338), and (337).
i} 0 =
Wr4+m-19 and V U,=Urs
n+l
344. Cor.1. If 7’u,=
: ui Gmn-1
then will (‘S., — a) NG tie— Gro ls
345. Cor. 2
n+
gu}
ca —1)?. (s Gna ‘) 0. GyUz= G;.A?.V4Us-r
n+1
346. Theorem.
For, G,.A,2u,=(t7!—1)". Gy. Us,
n+1 a
=S.(-1)7—*: m—1 EG bes
m—tI
n+1 mt
=G,.S,,(-1)""' : m—l
| m— 1
n+1 [7
. A2u,=Sn(-1)"" serie VR
|m—1
A2 U »=Sm (- 1)" —1 , m—1
. Urin—m+1 b)
[me
*Uz4+n—m+1°
|m—1
(341) ;
(86) and (6);
“Usin— m+19 (6), (340), (338), and (337) 5
(339).
139
n+1 ie
347. Theorem. Uy4,=Sn————n As"
ae
For, Gy. the:n=t-”. Gy. Ug, | (840);
=(142--1)". Ge,
Gy Pres =F —1)"-!.G,.u,, (86) and (6) ;
=G,.S,—m=— .A,*-!.tt,, (841), (838), and (337).
2
n
n+1
= m—1, Q0
CS sg Dae Ne ps (339):
| ma —1
~ |mtmr—-l
SAB. -Theoren (U2. = UF. Sp ee AN
| m
ry being any integer.
6
For, pubic 1. ra then
92145, = dns Sx" m.v1t, (221), (207) & (197);
n+mr—1
Se ae ete (196), (210), and (6);
nn
|m+mr—1
=1+4n. a ean ce
140
. |ztmr—1
Gy. Urpa= §14+2- Sa .t™ (1-1) G;.u,, (840) 5
m
| 2+mr—1
Dye ae APU, «its, (6) C2
m
(338), and (337).
ss |w+mr—1
. Mig n= Ug tN = [m HEN Pa ste (339).
m
349. T'heorem.
m—1
n+l 2 me
ris, Gey ey et ae
2n-2
|2m—1 7 Uy—m+1
m—1
n§, Bt)
tee | 2m—1
2m—2
Nene Tees
6 ne
For, let @ and 2 be any quantities less than unity; then
eee = > (12) and (9);
we
(plSBe 1
VTZOE 4 0
4 1-6¢
~ 1-0(614+2) 40
1—GF
-» where s=¢(¢-'—1)’;
me —9(2+2)+6
141
aes
~ (1=6)'-08
co (Pe ee
=(1-01).S. Gop (12);
Qn
=(1-62).8,6""!.2"-1.S,, a .6"-}, (92) and (39) ;
ie
|2—2m+2
20262). S'9". See —.s"", (16) and (26);
M—
Q2m
n
= 0t)-S,6° 1/9, -te e— (8).
n—m
| gm |m+2—1
—m,1 _ n—m F
Tene aaa OO
|m+n—-1
nm+m—i
ie eee =(1-00).8,07 Se
ih , [atm one |w+m—I
=S,,6"-!. ee tO aa apa (6) ;
" nar [EM , [tml
w! = =2m—1 m—1\ NMS | mind
=1+58,6". Sn = PS il Re ne t (9), (6).
ag WiaeLe 2 [etm —1
» f= 2m—1 gm-l_y S 2m—1
m—i
— 2 A me +
|2m—1 |2m—1
, (mtm—1
=S, gm (gt yem—2_G, Se g(t 1y"-2, (6).
|2m—1 |2m-1
ai | +m , \vt+m-l
2m—1 2m—2 2m—1 2m —2
» Ugen= oh ve : jis I abi ——_ A; Uz—m3
— m *
2m—1t |2m—1
(340), (342), (338), (337) and (339).
But |[w+m=|n+m.(m+1).,m—m+2, (41) and (40):
2m-1 m—1 m—1,1
m—1
=(n+1).P,§(m+m—r+1)(n—m+r+1)}
m—1
=(741) 2, }(m+1)’—(m—r)*t
=(n41).P, {(n41)'-"'t, (61).
m—1|
n+1 Pp n+1 ee
: Uryn=(N+1). pe at
2m—2
. fares Uy—m+t
op:
n n” —_7 02
—2.S3m (
2m—
“[am-1 ae Tae (6).
143
+1
+1
1 h 2 é ye m—1
350. Cor. Put Amb Cnaas where ¢,=Sn———. 2") 5
s °
2m—-1
1 1 Ch-1
then ei =Ch— te Gia: rep —C,-95
2)
2) -l
and fe =C,—C,-2+ (¢ —t)€y_\-
n—-l awm—1
vey 24 a
Now, €,—Cy-2= 9m [5 (|n+-m—|n+m—2)+2n.3" eat
| 2m—1 2m—1 2m—1
(9) and (5).
And, | +m = |z+m—2
2m—1 2m—1
=|n+m—2 $(m+m)(n+m—1)—(n—m)(n—m+1)}, (41);
2m—3
=2n.(2m—1).|n+m—2
2m—3
=2n.(2m—=1).|n+m—2.n.|n—m+2, (41) and (40);
m—2 m—2, 1
m—2
.(2m-1).P, J (n+m—1r—1)(n—m+r+1)}
m—2
=2n?.(2m—1).P,$n®?-(m-r-1)’t
m—2
=2n*.(2m—1).P,(n?-1°), 7 (81).
144
n—1 »~m—-lL m—2
2 Cg —Cqig=2N - Sn isoae .P,(n?—7?)4+20.2"-* 42"
=2n’.S,, ae P(r? -7") +2",
i n* =
since P.(v?-1")=n;
1 Le 5S =
Lane! ‘eae See ommee
ri Sn art r’)
n gm-l m—l|
ae 3) Ie es 2 me
+4274 (144) (t-1-1)-2-5n Spy he r’)
Ana (WUT ) m—1 =i am—-2, 1 =i 2
eae 2 : i =| mn 1. a =| en
Tee pee (Se GS)
m—l
n Pp n’ 9”) - 2
+n.S,, c (ape BF rae (6).
m—1
iy
wa (0? -7" :
Uppn=n?- Sm ees Ap Ug Oe eee
®
m—1
nm Pi =r’) bow,
+2. = Aer, cae ee eee
351. Theorem.
o m m-r+1
Ue =S, 8,9 | Wea parapet | OS
m+1
. s—l pss rh
where Vi Ug=Sr4,1-V" Unir—19 and b,-1=G@m—r+19 r=m+1)"
ll
m+) ¢
tg, i
For, put #=8,253 then a—z=—S,a,¢~", and
1 S,0°t=* = (12);
1
1—69¢-)’
145
1 S,a,0"-"-@".t7')
= Se . Ti...) aE
S.a,07-* a —6@". tz ")
=
eon, apeileme PE) (5) and (6);
ws
1 Ot S.a,0" "8" 35,a,¢7"
m ae 3 @r-}
0:0 9. op aa
ee) ee ame Oy;
Sig Qe —z@”
iP
=—, suppose.
Q PP
m+1
Then 55 8-1") (GC inn4107)-%, (12) and (8);
ag. sr : Gane ) ‘S, Ge-? sn i sla b, -1=4p- Br (eae =m+1
roy ay
=S.6*-1.S.s8-) gt BU bo;
m m—r+l Geen 7
a a — S
and) PSG" "5, race (22);
m
So-.5 mre, (gy.
P fP-}
-S ese S, m— al RA as= 1 ow" m(s—1)—1 es
a n . Yr .
&ly
ace gr- 1 ae G An - Gm= rt O81 yn=t—m—) 5-8, (18),
P fpr}
since a” a"=0, for every negative value of m;
Ah
146
ss 4 Im rtp 1 r+1—m/(s—1) 5
ea 25 3) Sa oS—1 gpn—rt+l—m(s— Sb
=
m 1 m—'+1 3 ’
Oren : Cie ane S el gt 1p t2—m(s—] Riss (22);
mn ] tl
=S, 7a ites So sl wo Sree Din: (8);
oc m esl m—rt1
=S, Sao Sp paw Oe 5) (Grandaqlige
m m—+1
x ree Sy Wee Uns y— ihe S por+p- 5 (OMoE pena bs; (345).
352. Theorem. G,(u,).G,(v,)=G,.G,(uz-V2)) U, being
the coefficient of s’, and v, of ?#”.
For, G,G,(u,v,) denotes a series of the form
— yO OO
=]
oN a
Sn S, An, n* s™ > b,
such that the coefficient of s*.¢’, shall be w,v,; and the product
—, 0 —%,o
(S,7,8") (0S, 0,0")
will be such a series.
. G,G,(u,0,)=G,(u,)- G:(,)-
353. (Corll. .s°™. 62" GG) «G,(0,) 9G, GAC as ten)
354. | Cors2.(s-'=1)" 1)": Git). G@,)
=G,.G;. {A2(a;). AZ (,)}-
355. Cor. 3.) (s-1¢-*=1)":G, (u,) .G,(0,) =G, G;-L,. (u,0;)-
147
356. Theorem.
im
n+1 RAO
7A (2%, v,) = er : Janes ren (u, Hees - 1) . peg a Uy
For, G,G;-A,"(%,2;)
=(Sapema—l) GG, (as) |Gz( 0); aM SoD ye
= §(s-!—1)+8-! (¢-!-1)}*. G,(u,). G,(v.)
= m—1| (sue a eae et (Gr) pe rn (eee yaaa CU.) 5 (86);
ee GCN ro Uae) Gz iae Use (Oy ang 342)r.
ln
n+ 1 ped
=G6,.6,.5,—=— =a HAVE ep. BE (eae 1): A, ‘ng rv ib) (352), (338) and
(337).
Bry ales
- A?(4,0;)= S,, emals | Ges Be be a je Ne 1, Vio (339).
[mat
357. The symbol G,;.u,,, will be used to denote the
0, co — 2, &
double series $,. 5,8’. t¥.u
390: (Come Ie) 8”. bp. Gg eg Gag lo tm pin:
359. Cor: Gri Sn) Cea Gert SG a Ae Ny
to
360. Con. 3, (6 f *=1)*-G. uu, —Gs1- Ay ng
148
361. Theorem.
m+1n+1 | me a
1 —] | =
ao APA Uy
As): q pl 7-1
For, Gs4-Usemiytn
= fits 2-1$"$14t-7—-14". Getta ys (358)5
m+l1 | m n+1 We
Ol (san SS a 1) Ge 86);
= ( ) oS ( ) ot »Y? ( )
m+1 +1 |m [mn
EY CE ep Ae aay alot ar (0) We (09 (2.813
fey eee
(338), (337),
m+1 n+l | m f |r
pa) q—l —1 q-}
SNRs et adh stats
. Tis nie). Sy
|p-1 |q-1
and (339).
362. Theorem.
iz
a“ -Un4n—m+),y+n—mt1"
n+1
A’ ares) (ast ha i EE
_ n
For ’ Gs : A, yMay
=(s—125'—-1)*..Gp its 9> (360) ;
|
(Gian aed : Gags (86) ;
n+1i
=§,,(-1)"). =.
|m—1
n+l |
=S,,(-1)""1. ae G,¢-Usin—m¥tyen—at (6) and 4358)
n
(338), (337)
n+1
2 Urtn—m+l,yt+n—m+ 19
As Uz, y= S, ( = 1) -1, Ea
and (339).
NOTES AND ADDITIONS.
Page 10.
25,1. Tueorem. If a,_;=0n-1—Gn>
n
then a=S,,(-1)"7)-0n-1+(-1)" Qn:
Bars (a0) s) a (=) we One (ab) as
n
. S. (1)F 1-44 Sa (SI) baad)" -Ons (4) & (53) 5
n—1
n—l n = f
and y+ Sn (—1)" dn =Sp, (—1)"72 "1+ Sy (= 1)" n+ (= 1)" Ga
(9).
n
oe Gy) =Sn(—1)"7) -Om—1+ (- 1)”. On
cancelling identical terms.
Page 16.
42,1. Cor. This theorem being true for every value of
a, and m, we may put a=0, and m=—1, and we shall have
|0 =1; which result will be found of perpetual occurrence.
150
Page 18.
Ap. "Theorem: Siti ne — Oe ie
@,,=n anda, ,,,;=0, then shalt
For, Ay 41,m+1= yy, m+1t Gy, m+
n—1
=
Gy ea ne (24) b)
=S sinc =0
=9;G;,m9 SINCE A) m+1=9-
Hence, putting m=1,
SW Sart
Ti aaeP (48).
Also, putting m=2,
ma nN
i= >) 2-85
Suppose, therefore, @,, .=——>;
n—-1 n—-1 ig |m
e)
then Ay, m= Dr Oy m= Ore =-— (48).
If, then, the law were true for m it would be so for m+1; but
it is true for 3 and therefore for m.
48,2. Similarly it may be shewn that if
Ont 1,m=On,m+4n,m—-19 Uo=1, G,,;=1, and a,,,,,=0, then shall
\m
m—)
Se ee
nym—) :
|m—1
Page 20.
m,n
54,1. Problem. To find the number of terms in C,a,.
Put b,,,,=the number sought ; then
Ort aed Ont Cains (54), Dee and OO.
|
“6 b,, Jet (48,1).
m
OG
Page 22.
60,1. Cor. 2. Ife is independent of s,
m,n—-—mMm Mm, n—Mm
then, ©7 2 \(a,)\(b.c), —c"-".C, (,6,)-
Seew Art. 55:
Page 28.
74. The theorem of this Article may also be proved as
follows :
It will be readily seen that we may assume,
n+l
|@+b=Sy6n,m—1- ge ee ?
nr n—m+l,r m—l,r
where ¢,,—, 18 Independent of a and 6;
then |a+b=(a+b+nr).|a+b,
n+1,7r nr
n+2
or roe ONE as |b
n—m+2,r m—l,7
n+l s aa
=SnCgm-i-[@ - |b (a@+n—m+1.7+b+m—1.7).
n—m+1,7 m—l,r
P52
n-+-1
1
. Cn+1,0° |@ +9); Chaat \@ . (oe
nt+l,7r u—m+l,r m7
1
Bice, (ja : [jo + | Geek 1)
n—m+2,7 m—l,r n—m+l,r mr
n
é | ‘ s
=Ch,0° a +Sn KG) mp Hama |a °: |b +6y,n+|0 3
n+1,7 n—m+l,7 mr nt+l,r¢
*. €n41,m= Cn, m+ Cn, m—1 3
also we have ¢,,=1,"€; ;=1, and ¢,.,,—0-
in
aie OL Gee (A602)
|m—1
The theorems in Articles 86 and 161 may be proved in the
same way.
=} am} n (=a) am}
83,1. Cor. 7. (S. la. ee) =e | ma . >
m—1,7
for every rational value of 7.
Page OF:
re ™
o “——) © (20) =. ae
EE SS ee
102.1. Cor. 6,1. (S:
|m—1 |m—1
n being any rational quantity.
Page 61.
17%. -Or thus:
Since, D,~' HD Bice (ED ee aa.
=D,"""'(u).D,-" E,""'0-D,-"}. D,"(u).D,-"E,"0t, (176);
fi & Due) =5, (1) 2. DPA). DE aes)". D,-}
DQ). DD, Eto) tse (25,1).
153
Page 52
152). Theorem. D," |@ = | m. live |e :
. m,—h n m—n, —h
Bor 2 |w = |o+h - | @
m,—-h m,—h m, —h
=(@+h)|e —- | w (w—mh+h}
m—l,—h m—1, —/h
=mh. |v :
m—1,—h
D?\ «x =mh.(m—1).h. | a
m,—h m—2, —h
= |m Alike keh
one m—2, —h
Similarly, D,” ja = | me. h’. |w
m,—h n m—n, —h
Page 53.
ea | a D2 u
155,1.. Theorem. Soe al : 7 ; where h=Dze.
m—l ra
ao
For, put u=S, |v -Gmn— 15 Where @,-, is some finite
m—1,—h
quantity independent of v; then
D;"u=S8,, |m—1 "|v > Qn-19 (136), (137), and (152,1) ;
n—l m—n, —h
n—-1 :
=S,,|m—-1 hn. | RU eae Cn Rw Ra
n—l m—n, —h
=
+S, nm+m—1.h"-".| a AC ey sy SEO) and (42) ;
n—1 m, —he
(3)
154
= baihinw @. =
oOo
+S, |2+m—-1 Boa : .|a@ *An+m-19 since | 7 — 1=0, Co .
a—l m,—h n—|
«=0 =
1 1 ae
and @n-1= m—-1 ° hr-}
&
m—1
= m—l,—h Deas U
- U ‘m a4
ih
Page 71.
n;™m
207,. Theorem. d, Pom «=P, i277"): Gined: tg» Pt, >
a, being independent of «.
GOP ia Fd ain”
For, — =Sn » (205)
P,u," U am
n a,,d,U
=S,, “> (207)
m
ze = m ie
d, Pa ees) ee i a Gy
Mm
nym
Sp, Cn) Sua d,Um» PU, -
Page 73.
213. This theorem may also be deduced as follows:
wv
2 Da a
We have u=G,, meat, 0, where h=Dzx, (155,1).
|m—1 eas
co gm 1 DES 1 Uw
then w7=S,, —— - ( = )
h=0
ic=)
= Se
idez.u, (188):
Page 76.
222. Or thus:
Since [.§d,"-!u. f"-*v} =d,"—1u. fmo—f5d."u. f2"0} , (203).
2 f.Qur)=S,(- 1)" dee. for (-1)". fede f2"?)s (25,1).
Page 94.
253. The symbols ie u, fu, and ([,-f,_,)% are equi-
=a rth
valent to fw—f,_,U-
Page 100.
270. Or thus:
: tan v+tan h
Since an c+h)—ta w= —
i—tan wv. tanh
tan h. (sec «)”
i. 1—tana.tanh-
tan (vw+h)—tan &
. G,stan » fee
h h=0
7) A p2tan @ tan h
tan h (sec x)”
h=6
=(sec v7)’, (266).
156
Page 100.
2703. Cor. 1.,.d-cotg=d-Atanz)
(-1).(tan w)~*. (sec v)*
sec 2” :
=) ) = —(sin v)~*
\tan &@
Il
—(cosec wv)’.
271. Theorem. d,.cosec x=—cosec «.cot v.
For, d,.cosec v=d,.(sin a)~
=(—1).(sin v)~*.cos a
il COS &
sin @ sin #&
= —Cosece wv. cot av.
Page 105.
281. Every term in this series will vanish in which zero
is a factor of |m—m , that is of
m—2
(2—m) (m—m—1) ... (7 —2m +4) (N—-2m+8).
n
(1) Let 2 be even; then »-2m+4=0 if m= — +2; and
n—(4n+1+7) = |hn—1-r=0, if r>dn-l.
$n+1+7r—2 tn—l+r
Lay. eee
(2) Let 2 be odd; then »—2m+3=0, if m= —— +1; and
~
|r—.h(mtitrfa|s (anise? p) if rp>3(n-1).
$ (n+1)+r—2 + (n—1)+r—
Hence, the number of significant terms will be tn+, or
3 (m+1), according as 7 is even or odd.
157
It will also be observed that, although » appears as a mul-
tiplier of the whole series, yet the coefficient of the first term
n—1
all
a
[on
4.(2cos 2)".
being , (48) and (42,1), the first term will become
Page 116.
300.1. Theorem.
log - =2nlog 44+ 4log |2—2log 2n—log (22+1), (n=¢ )-
For, sin v-0.P,| (: +=} (1 -~) \ (293);
rT UD ULS
(27+1)(27r-1)
Ag? ¥
|
wo / 3
ofl
2
Te == 47°
Q
a(n)
Tq)’ (n= );
n,2
9
but |! = L zs :
na 2. |
a 4".(|2)?.4"(| 2)"
2° ((2n)y?.(@n+1)
a. ((n)'
i (|2n)’. (2n+1)_
~ log
y
oO
=2nlog 4+4log |2—2log lgn—log (2n2+1), (n=0).
0/3
158
Page 117.
301. That there are not more than » different values of
2Qm—1
E€ 7”
a may be shewn as follows:
For m put nr+m, when 7 is any integer and m<n; then
2m—1 = 2nr+2m—1 om=1 ae
——.7V-] LIS LLB ae Pe =t
€ becomes «€ ” = e2™N-l gn
=e” > (288)
Page 118.
304. For m put mr+m, when r is any integer and m<n;
ida also se 0
=¢@2r7™V-1 .¢@ 2
Qmn—2 =
Se 6 VI
then € becomes « ”
306,1.. Theorem.
n
x" —2cos 0.2"4+1=P,, (a -22.cos
For, put #°”—2cos 0.#"+1=0;
then a”=cos 0£+/ —1.sin 6
=cos (m—-1.274+0)£/ —1.sin (m—1.27+46),
m—1.27+0 — . m—-1.27+0
By (=), (77).
m=n
and «=cos
-, v"—2cos 0.u"+1
a m—1.27+0 ; —1.27+0
ee fe (cos Sa eEN ey —1.siIn .———_———_ ] [X
n v1)
m—1.27+4+60 — =. m—1.2c+0
v— C08 in Od in pe
n
a m—1.27+0
=P, (@’-2 v.coSs ——__—_——_ ])..
i]
159
Page 132.
327.1. Theorem.
va (tan di) eee
(tan w)"= =) (1) damaee
(tan a)=S,,(—1y"-". SP (ay (tan 2)
For, (tan x)"=(tan «)"-*. (sec #1)
= (sec #)?.(tan 2)"-*—(tan a)"*
r
-. (tan 2)’=(sec x)?.S,,(—1)"7. (tan x)*-?"+(-1)". (tan ay'-",
(51) and (6).
” tang)" 2+! ¥ a
AON f,(tan a=-5,(-n +(-1)". {(tanz)*-*, (270).
327,2. Cor. 1.
(tan ti) ita +1
yn
[(tan x)*=§,,(-1)""?. ray a +(-1)*".2, or
$(n—1) tan v n—2m+1
=S,,(-1)"". a +(-1)'" log, cosa;
according as 72 is an even or an odd positive integer.
327.3. Cor. 2.
ry
(cot 2)" =(cosee @)?.S,,(—1)""". (cot 2)" +(—1)".(cot a)'-".
r cot & n—2m+1
(cot 2)" =S.0—!) ee +(-1)’. f,(cot 2)"-*", (270,1).
160
cot x n—2m+1
(cot ayant
in
J,(cot @)"=Sm(-1)". (-1)*".a, or
n—-2mM+1
+(n-1) cot &@ n—2m+1 :
=$,(-y7, ao 4(—1)**™ log, sin v5
according as ” is an even or an odd positive integer.
ART.
10.
11.
INDEX TO THE THEOREMS.
n n
Ir an= Dn 3 C=); then Sn hyn= Sn Dn >
Si (4, Li Dm) = Sn ant Da Dn :
If b is independent of m, then S a, 0=0 Sin Gn
S,,b=nb.
n n
Sn a= SU eee °
n r Corer is
Sia On = Sn Qn, Tr Shi a, +m*
S m —1 Lew,
CO
1-@
a"—a“" kn
= ert ate}
a-—x
1 We th
Ppa. nae , and
—_—¢ —Vv
1 n a”
FS | (a ly eae PR nape (al he
1+ nf ) ( ) 14+”
1 o=)
If «<1, then ee
u
1 co
apt — (leo) ant,
1+
xX
PAGE
9
16.
Wc
18.
19.
n
NM
if a,=0, for m>n, then Sra Dy, Ola
2n n
Dy an= Sn Aon = 1 + S, om =)
2n—1
S., an = = S,, Gam -1 a5 S, Coy, 5) and
S cp
= mle Si
S
(Sp tt) x (S, b 6 a) S,0, *:
oo
If r is independent of m, and s of m,
r § so
then Sn S., On.n= 5; oF Gn, n°
=] m
Sn S., Qn, n= Da na Lin —n+1,n*
m
co
Y SLO aS Sis heh ae 1,n°*
$28
©
S,,a
m n Gn, n— /m Qn, +n,”
a m r
cols:
Sn Sn Gn =e Sr An - rcs nm +7—n+1,n°*
>
m m—nr+l
m n
S, Sa, Sie aes
n m—n+l1
and =S) S,.4a Antr—-1, 0°
2n n mn
S., S, An, y= oh S, (d.,2 1, rt Bon = Met rte T+1r+1
m—-1 m—n
+S, S, (Gorn ip); rane omreiprtade)> and
2m—-1 n m—-1 n m—1
N NM
S,, S, a, Se Ss, S, (on —-7',; rt Aon — r+1, ») ar Oh -—7’—-1r+1
m—2 m—n—1 2Qn—1 .
+S,, S, (GigE ant Corea tad ee Qom 1, r°
PAGE
or
INDEX, 163
ART. PAGE
n—\
OA. VE Gai=GatOny (en2,)> then’ @,=a;F Os 5n- 10
n-1
1); then?a,=e'r "a, fa Oy 5-
m=1
m=n —
oe TE Ga 6Ge bn. (
= m=1
2551. iif Bm —= On —1—Ans ( )
m=n
n
then a=S,,(-1)"7!.0n_1:+(-1)"@,- 149
co <<)
NX n—1 n—1 re m—-1 @ n—1
20. (S,, Gn Bree ) (Seba Ssh An — yo ek Ad er ets
eo m
a m—1
=Dn0 aa Oana: il
co ©. m
x maI\e Sy amt @
21 Aah UPC OR Ua) kV Rs AY a?
n n
m=)
29. If ay, = b, ’ ( ye then | og ay = | 2s Dn, : 12
mn
nr
30. If bis independent of m, then P,,(a,6)=b". Pn dn-
n n
oO
31. RG RG Pema 13
n r nr
32. m Cyn = PS (4,,) . m y+ m*
r n n+r
33. P (Gm) 0 P,. ay 4 fa Po Ay, .
0
34, Pya@n=1
—n 1 1
30. m&in= Hh er
Py Ay —n m a_ (m—1)
n—1
86. If da 41=,n-0,,5 then.a,—a,. P.,8,,- 14
. n—| n—l
acy Lin ar +2=4m- Dns then Cry = AQ - Nd Dems and an 1=Q- P,, Bom —1-
39. |ab=b". |a 15
n,m m.
164 INDEX.
ART. PAGE
40. |a =|ain—1.m. 16
n,m N,—m
41. |@ =|@ .|a+rm.
n,m T,m R—T,M
42 | @ =
0,m
42,1. |O=1. 149
1 1
43. oS SS = SS 16
as |a—nm a—m
mee n,—m
I
44, ——=0. 17
—m
jn [w
46. —= —
m | —m
EA ass 2 ee
ANT m i m+1 a m+1 18
| m |m+1 m-+1
48. If m and m are positive integers, then
ie a [2
m+] m ™m oe s
wt 8, and — is an integer.
[mer |m? [im :
4851 If An+1,m+1=4n, m+i1 tn, mo Ay, =N, and Qn, n+r=95
In
then @,,,=——.- 150
|m
48,2 If An4+1,m=4n, m+ En, m—-19 Ay, o=1, @,,=1, and Gnjn+v=O5
|
m—1 a
then 0.5.85 —
| am a
INDEX.
165
ART. PAGE
m—1
50. Jy Dy ee Ce S, @. |. 0, Ke: P, b;. 19
m+1 n+l
oe
If On=0,+C,+Ons a>
s—l
then 4,=8, Geese Prone at Tn+ma: pret l)a
52, If On —Un gO n? On+a,n+B then
r s=1
as
Ss One Gotan tC 4* een) aoe 1)8
a
On eraarrb tenet ae n+ (t-1)6
19
m+1,n+1l m+I1,n My, 1
54. C,a,=C,a,+4,,,-C,a,-
| 7
m,n a
54,1. The number of terms in C,a, is—
—— 151
[m
mn
55. If 6 is independent of r, then C,(a,b)=b” C a
56.
ie
If a is independent of 7, then C, (a)= —.a". aT
57. ac
58. C,a,=1.
60. If b,=6, then C,,,(4,.b,) =b"-".C,a
rhe
60,1. If ¢ is independent of s, then
mM,n—m
m,n—mMm
C,,5§(4,) (b,c) } =e". CG, «(a,-8,)-
n—m+1,n
n—mM,™m n—m+1,m—1
61. C,,s(@,-b,)=Gn41- CG, s(@,-05) +0, 41: C,,s(4,- b,).
151
(h3)
tw
fae
74.
INDEX.
A i: ee ps Lae PAGE
bl, =n On Af eee & AG 23
0
A.,,@,= ike
n na
If 4,=Sn@,—-m-On> then a,=d,.A,,b,.
n
if a6 2S 2¢,2 22055
n m—1
thenta,— One ee senes Fae Q4
n+1 n—m+1,m—1
P,(a,+6, )= =h)5 C,,;:(@,-0,). 25
n+l n—m+l1,n
P, (7+¢,)=S,,2"-!.C,a,. 26
If a, is the r™ root of the equation
n+l n—m+1,n
0=S,@n-1-0"~', then shall a,_,=C,(—a,).
n n—r,n;t
oa (- ao
Pig —A,)
If b5=S,0," 5 2a-e(C-), them =
at / b—a,
If pf oaecua! eas ( =); then x=P, ( ) : Bil,
a,—d,
1 n+1
S a ym) n n—m+1 Sinn =e cite
a ea are = Oi. ee
—b «x—b
n+l
If b is a root of the equation 0=S,,a,,_,.@"~1, then
the second side of the equation is divisible by #—6.
| m
n+l aes
a+b=S,,——— .|@ |b F 28 & 151
Nyt m—1 n—m+l,r m—l,r
nN
n+)
|¢—-b=8,,(-1)"". ee : oer 29
nyt n—m+1,7 m—l,—r
| a+b aa |« : |?
——— ees
nr = = 30° —1, +r
U eS =S,,(+1)" a n—m+1,7 m +9 .
| n—-m+1.
INDEX. 167
ART. PAGE
|b at! |a+b.0"-
co
|m—1 (pal |m—1
a an} | ma a ee
=) a ==) ’
—l,r m—l,7 : =i)
78. ( aa =S,———-;, n being any positive
F |m—1 oie ea ee a
integer
79.
Hin
as
i
=)
8
3
vi
“—_-
|
8
&
ms}
|
3.
=
81.
t
oO a a 3i\n 2 t gn!
on laue a, lat east $1
83.
(
(
(
(
; 2 rational. 152
(s a am} n © na am}
mache | 1 ce
t m —1 | aan 1
84. (Sars. = =) (Sie n n—l,7° ee |
1
oe It eee [m1 ae gas in. 31
* Stainville.
83,1.
86.
88.
89.
90.
|m—1
as 2 is even or odd.
nv
(a- —b)" 26 Lee oe (aby? (a
n
+(-1)". ES (ab)*”,
2
+ (n+1) WG
or =S, (-)* 7). md
[m=
INDEX.
1 , PAGE
t a an n t na gr ;
(8.55.7) =Sn 55: [m—i + terms in 2. 32
: | m— , m—
n+1 | 7
Sie Sp -a"-™*1>"-); nm a positive integer.*
a b)” n+l qr-mti 67-2
CD eee i
| 7 |~—m-+1 ; |m—1
é in
(a+by'=S, == (GB a Bh ie .(ab)?",
ee 2”
2 (n+1) 2
of =Sn———.(ab)"* (a? 4 bm +2) = according
Click eal ier ar fy
.(ab)”"- ‘a —2m+2 oni ae OS
according as 7 is even or odd. 34
#7 co nm am} vas Y
Gp) =S. p=—. ; mand r positive integers,
7 |m—1
m—1
and #<1. .
rE a
(i+a)'=S,——.a”—!; » rational, and v<1. 35
* Newton’s Binomial.
ART.
93.
OA.
95.
96.
98.
99.
100.
101.
I
3
ry
5
[oF
INDEX. 169
PAGE
n [m a\™ 1
l+v)'=S,—. (=) 35
(a2) oes r 2
ae a |n—1 I v m
1 au 1 m— ae m—l, m1, 7n_ . (=) ™
Gray S.C0 eee
=
Oye,
l+ax 25 1 m — Lf min n / (=) 36
oa Reise.
2 at lan) b ] a” 1 (a+b)”— 1 am 1
(see Cue ac
m—1 |2-1 m—1
n © gm Lym-l Shi: m—1 am 1
P, (ie “——} = BS Msg
|m—1 |m—1
oo a™— lan} n © (0) Pca a
( ea =n a ; m a positive integer.
co a”™— 1 gyn 1 oo m—1 gt}
SS ay aie -8, (< ) : ary
a” ly Os: 1 a} -) (a—b)2- 0 m—1
ce io - 1 a (S, [2-1 1 —~——|=S. |m—1
a” Cores! 1 ee 1
(s m rani. 37
|m—1
pee © (2p4)" 7 sa
|m—1
) a®™—), am} nm © Ge)"
102,1. ——— —_—___—
104.
105.
106.
107.
108.
109.
INDEX.
; 2 rational.
=m
S.$ Sa
R=) at} n 2 ‘an m—1 .
(Ss .) = ee n rational.
2 (0/ £1)"
v= Sn et
; a rational.
€ ; w irrrational.
| m—1
ym —1
a
| m =
—_—
c
€ = Sh
; @ any quantity.
1)
fe a (w.log, a)"~
m—1
(a£b)?=a?£2ab+b’.
n—m
(Sudp)?=Snt2+2.Sylin- Sines:
eow=e"),
p'(u)=w.
pr-gr(u)=.
IE pale, qpr)s pale), &e. and aCe), Pale)» Yale),
are all distributive functions, and commutative with
each other, then shall
n+l n—m+1, m—
(p+) ; u= Sis 5 ey s (W, o 7) t U.
r+
37
38
4.0
&c.
Al
ART.
130.
INDEX. 7
PAGE
n+1 ia
( +W)n Uu= =a n— ein ai '(w). 43
p =e
(P+W).=(P+y)r-u
If W(w) ss Sa Qn, —-l° On i(%), then W"(w) a (S,, an— 1 @n- 1)” U.
If W(u)=S,,0" "gp"? Us then wW"(w) = Qa —a.p)~".u
If @, denotes such an operation, performed with re-
spect to #, that b,(uw+a)=,(w); « being independent
of x, then shall
n
Pz" , p(w) =U a5 S,, pr ae bs Cm 3
where ¢,, is some quantity independent of wv, and is to
be determined, in any proposed case, by the conditions
of the problem. 4A
Du=E,u-u, E,u=u+D,u, and u=H,u—D,u. 45
E,-9(«)=9$(#+ Da), D,p(2)=(#+ Dx) -(2), and
A, $(7)=$(v+1)-(2).
D2 t= Do 1 Dea Ek Pe
Efu=u, and £7". E2u=u.
E,.p(«)=9(e+nDa). 46
dds : (2) = plwv —Dzx).
E,”.o(@)=p(a-n De).
E," (u+v)=E, u+E,"v.
* With the same limitations as in 118.
172
ART.
135.
136.
137.
138.
144.
145.
146.
148,
149.
150,
INDEX.
PAGE
E,;" (au)=a.E,"'u; a being independent of 2. 47
D,(ut+v)=D,u4+ D,v.
D,(au)=a. D,u.
D,(u+a)=D,u, and
D,-"D,Zu= ei ID: ~@-™ 11, c¢, being independent
of 2. 48
D,"(u+v)=D, 'u+D, 'v.
Da) =o. tt
E,.D,u=D, Eu.
E,.D,7u=D,7°E,u, D,E,'u=E,7!D,u, and
Ee De a= DE. 49
n+1 [a
D;u=S,,(- 1) j peeled i Bie m+lay.
m—
an+1 |
EZu=S, peed ae _D*- ly
"maa
a" = | 7. (Dx)". | 50
n+1 ed
A? .=S/(-1" 4 ie .(w@+n—r4+1)”.
n+l m
S(Sn)it. LI = (w+n—-r+1)"=|n.
n+1 [%
A” .0"=S,(-1)’-). —— . (m-r+1)”.
ata
INDEX. 173
ART. PAGE
151. D,?.a"=a".(a'*—1)". 52
152: a+bxe=bnh.|a+b.(a+h).
ny bh n—1,bh
154. D,(|a+ba “l=-bnh.(ja+ba)"'. 53
n,bh
n+1,bh
Weise, Ae
156. D,.P, Sv+(r—-1)h} =$p(w+nh)—-P(«) Pd (a+r).
157. D,[P,o a+ (r-1) ht]
=—So(x+nh)-9(2)} [P, pie+(r-1I)ht |. 54
158. D,.P,(u,)=S, Cyr (tts Derm):
159. D,(uv)=u.D,v0+D,(u). £0.
1600; . De. bs as D,(u).v—u.D,v 7
: v0.0
pees oa
aor ea lee! a ey ame)
where o(w), di(w), (v), and yy,(v) are distributive
functions, commutative with each other and with a
constant factor.
162. (PW+gi),.wv=(Pyr+ giv)’ ur. 56
174
ART.
163.
164.
105.
Or
166.
167.
DPP, (u,)=§(4'D,)(142D,) 02. (14"D,)— 2". 10 2la. 0. Up
168.
109.
170.
(le
172.
174.
176.
INDEX.
(Spy, wr= (S, pry," ur.
PAGE
57
s m
(Sor 2 SPr)nUi Us ~ 3. (S20, Oe 5 Py)" Uy Ug. U;
iD? (wv)= ie 5 BES 1(u). D*- m+1 oe Igy
cer
n+l |n
—
D,?(uv)=Sn(- Mera: Tet Sy) Heal (73 al Obici
Dj (uu) = §(1+1D,)(1+?D,)-1?" a, u,, and
E,E,ju=h,E,uU.
E,E,u=u+D,u+D,u+D,D,u.
D,D,u=D,D,u.
Ej. w=(1+D,+D,+D,D,)"u.
D,” u=5(1+D,)+D,)-1}"u.
DY
n+1 mn”
a Sn(- yee =. imo iy hae 5 eee
n+1 |
E} a= a ee rr
\m—1
oi s—l —s§ s—
Ee Ej u= Se ape ae ‘D, lay.
D,7\(uv)=u.D,u-D, 15. D,(u).D, Ev}.
Dix (wv) =, (-1)"-?. Da (u) Bi) Paps: Oe v
58
59
60
+(=1)-D,-{ D2 (u). DE}. 61 & 152
ART.
178.
D,-"(Uv)=Sm oe ese 1 Be (w) Die py Se Vay,
M—
179.
180.
181.
182.
183.
184.
185.
186.
187.
189.
190.
INDEX.
If, for some value of r, D,’u=0, then
n
ve
Da
7 +const.
Gl
nr
Daa a) (a. ln? se) ae on.
D,-'.(a*u)=a".Sp(—1)"=}.a"—", (a*=1)-*. D™y
+(=1)"D, "3D, (u).a2*”. (a’—1)-"t
|@+b.(w—h)
D “tlatba Se
ae bh(n+1)
= Aci
D 1 1
“|a+ba — bh(n—1) |a+ba
n,bh n—1,bh
nm or | —1
ieee i) (n—-1)|a—-1
Se
+1 TINE IN Dic e+ ce
v= "|ma1 Vs and AW .a=S,, Shi oie
mM m—l m
n
=| =i
Sn an= (A n (AVE) An+1°
a,” .d“u=d,"* a.
dQu=uUu.
03
64
66
176
ART.
191.
192.
198.
199.
INDEX.
PAGE
d,.(w+v)=d,u+d,v. 66
d,.(ut+a)=d,u; a being independent of «. 67
d,:a=0.
[tdgu=u+S, eee
d,.(au)=ad,u.
dn.
p(e+h)= LS, ae ds"). b(a)* 68
2 pm a m
Rae 2° u, and Dyu= Sat A.
Dfu=(e'-1)"u.
d,”.@(«) is the coefficient of = in the expansion
of d(v+h). 69
If w is a function of v, then d,.p(u)=d.p(u).d, wu.
d,(uv)=vd,u+ud,v, and f,uv=u f,v—-f, (d,u. fv).
d, (uv) x du iM dv
Uv U v
ast P, Mt, <2
NBL see ae 70
nay Un
n n nsm
d,-P,,=S, Fem Py Uy:
d,.u"=nu"—'d,u; mn rational.
* Taylor’s Theorem.
INDEX! vies
ART. ’ PAGE
“uw ufd,w dw hi
208. d,.—=—{—— - = iI
v v7) U v
m ™
P.u, Pu, ce dell x dv
209. d,. ” —< = n == {S. =P ar
Whe
Pye, P,»,; ‘
Ow jo, Ute, n— WL
910. da =|n 4 ;
m
4 ie
D4]. Fike (wv) = Se ™m— 0 mie C ais Tay 72
cea
m m m
212.9" 0 Pa. = (S,"d,)"2 Prt,.
218. If wis such a function of w as may be expanded in
positive and integral powers of x, then shall
214, d,d,u=d,d,w; v and y independent,
g8on dud, dU, 74
m—) m—1
1 ieee
216. P(ev+h,y+k)= Se Se ee ies? od)
m—-1. [m—1
m hm-* : ke-}
BUT. wo) (wth, yt+h)=Sn S rifle ices a p (x,y):
|m—n ; [2-1
218, If w is such a function of a and y that it may be
expanded in positive integral powers of v and y, then
shall
(-) © an} Te
a (pew:
“|m-1 nl ve. es
* Maelaurin’s Theorem.
Z
178 7 INDEX,
221,
225.
226.
231.
232.
If visa sare of w and y, then
ds. }d,z-p(2)§ =dyhd.z.()}
If y=\)}x+a.p(y)}, where x is ee of &
and y; then shall
FU)=AY@) +S. rm ds" " ow.(2)|".d.fy-(x)}*
If y=x+a-p(y), then
IW=fO Sai d."—".$(s)]".d,.f(2)} 4
as) 8, (1) dn. noe (1) Gana.
PAGE
75
76 & 155
sm
i a= Bi(e Ly" 5 GDA rea
mys S n+1 m—1,t—1
S,, eC a= =§,"- et ees ny (ae) =
1 m m3 a,s
Giang.
1 m m,n a. U,
[rn Bo PP; U, a +s Ae: 2
0
w,?.p(a)=$(a).
m m—-?r T
Se [oR ty ° -
w”".a’=S, | 7." op Teehnal-
a, oe |7
[n
m
mw”. ae CP BAS Sides Eo
If 7 is a positive integer, then shall
wo” .a” m ie " ont a ae m as m+r—l apr SUE ie
=S8,
" iapieos s Theorem. + Lagrange’s Theorem.
Ey [n= [rs |r cs Eee eg |~— m+r—1.|m— c= ia ee
80
81
ART.
242.
INDEX
ppt a N c,n,m a’
|m
r—1
=e +§,+5("—1)
&
me —a,\
COO Nas ( } .
a
aa, = n—1
is Am — 1-0” ‘ =p
$$ $$ = ym—1 QS / t
s i =~ ,,0" Ss. Ft Rarer Nee ( j F
a ee
S,, Do 1¢ x”
a: = 1," P(u) (2) n wm =m gym d may
4
cae =5,,d,".p(w). ; where @,,_,=
im
=)
p (S,, An — ya" in ')
oe) n aw"! ay we
a (a)+S8,2 rS,,d dat a abate
? 24 pKa) |n —m+1
1 x 2m ail
2 aa 2m—1 me eet:
e- =P 9 Sind hal rt ).
2n+1 =f
peer
\ r+i
1 =
7 =@-'-348,,6n-).0"-).
e —l
-
= = ~ 9 é
DD, u=h f0— = ASp Cam 1-12 hme
2
tO
Nee 2m—1
a. w= fu age +3n Con d, Tate
a” +1 i y”
AW .a= a
2+) 2
2m—1
d,.e"=e".
2 +5n Gon- 1° |n Oe a +14 const.
179
PAGE
85
86
87
88
8)
254.
250.
INDEX.
d.” .€ on €
ad.6 =e .04
ae : = « 2 . (ee
d,'.a°=a". (log, a)’
1
d,. (log. v)=—.
i
d,
d,. (log, “) = =
Ly du
IO a u
d,. (log, w)= lo
PAGE
Y3
94
pes A ym — Ca —
log, B= Far dak 1) (= 1)’. eae
| Cas (a"—
If a®~1<1, log.2=—.§, (-1)""}.
re m
ym
* : es a
it a <A, log, G4+0)=5,(-1)" at
m
<<) aL
x « $
log, 1-#)=—-§,,—, and
mm
1 1+@ = Lo
Oe {—« cine Qn ay
] © qn ae 1
log, v= —.S, ————- -
fie ideal Maen ie)
ye} >
ple’ )= Pees [m—1" PU +A)j{ om-!.%
gitm—1
BIN? o”"+ m—t}
Nese ae n+m—1
=1
+\m
Ail
See 95
96
* Herschel’s Theorem.
ART.
259.
PAO
206.
267.
268.
INDEX, 181
PAGE
1 * an} m A®7);077
=S, = 1) 7) 97
e’+1 Tani n(-1) gn
r ee yn} & (<1): At) .0° =
at (sa may j n
a v co yaa ae (Xf>1 Qo?”
—— =1--4+§,,— -S, (-1)""!. ———_.. 98
e’-1 2 2m.
2 JON) )o3 IE
DU=Sn ee eae
n+m—1
jlog. (1+A)}".0"=0, (m=); and
Hog, (1+ A) }".0"= |.
jlog, (1+A,)} w=d,u.
d,"u= Slog, (1+A,)} "wu. 99
sin 2 tan @
( =1, and =
\ v Pr Asef av r=0
d,.sIn ©=COS @.
d,.coS v=—sin av. 100
gq . sin cy -COS vy ae sin L= ( — 1a i sin av.
d,”"-'.cos v=(-1)”-sin vz, and d.2".cos a=
d,.tan v=(sec x)’.
d,.cot a=—(cosec x)’.
d,.sec v=sec x. tan wv.
(-1)’. cos a.
100 & 155
150
100
182 INDEX.
ART. PAGE
271,1 d,.cosec v=—cosec a.cot wv. 156
! 1
272. d,.sm-' #=———.. 100
J/1 — a"
: —1
273. d,.cos~' a= ———., 101
Af L—2e
1
974. d,.tan™
1+a~
Dies de sSeCr a
au*—]
n Ee. n Prats n
276. P,(cos v,++/ -1.sin v,)=Ccos §, a,+ / -1.sinS,a,. 102
277. (cosa+ / —1.sin v)"=cos na+ VY —1.sin mv; n rational.*
278. 2cosna=(cos a+ V/ —1.sin x)"+(cos e+ Vf —1.sin By.
105
24/ -1. sin na=(cos w+ 7 ean sin wv) "—(cos he Af 2iah sin x)".
a, o n—2m+2, 2m—2
279. cosS,2,=5, (-1)""- ~Goh} cos 2... sin ey,
n n—2m+1, 2m—1
Si S,2)— Sel) e C,.;— (cosa. sin ,):
= 2m—1, n
- S, (—1)"=".. Ge (tan 2
280. tan S,7,= Bat eae a
S32 (=1)"-.. GC, (fanz)
Aves n—m
281. cosnv7= tn. S.( 1) ee m2 . (2 cos ayn 2m 42, y way
ee rr
= 4 n. S,, (-1)""! Par (Qcosa atte 104 & 156
| mm =
* Demoivre’s Theorem.
INDEX:
ART. PAGE
P.
n+ (n® =: 47°)
A Fea eae aN SD te Ne ee \2m—2
282. cosnw=(—-1)*".n?.S, (-1)"7?. [2m—2 i(cos. a)" ~*,
105
P.§
be 4 (n+1) s n’—(27—1 2
or =(=1)) "2.5, (-1)" 7. [pm—1 ) Pan i) Hoe
an |
283. Sc ey a eae
m—
|
o-n in
+ : Ln’ or 107
lm
ee ee
an
3 (n+1)
=(—1)9) 2S, (1) a5 (2 297 +2)
1
tn [
——— cos (W—2mM+42)u4+2>". ES ;
(cos aay Sn [™—
|m
Ent) Le
or =2-74)..S- refs (n-2m+2) a. 109
110
oe
sin c= —] Bir
S,(-1)""). 5
e**V-l=cos ~w£4/ -1.sin a,
2 Gos @=e°V=14 6" V1 | and
2n/ —1.sin e=e*V-1—¢7*V-1,
290.
291.
292.
294.
207.
2098.
299.
INDEX.
PAGE
e2m—at VAT =(-1)"- 4/1, and e™)7V==(-1)""". 110.
R WEIS (> faa
tan =. (1 aS (s)- 111
an =5,(—1) S, |2n—2m-+1 PENT Bigeye
~
q on fe Sheen
m—1 ]
i yen 3 _— 9
cot P= S,(-1)"- S. = An, [) ) ries
294-1
o-) n—l =)
cosec t=§,,(—1)""!.a". A, taal
2r+1
= av?
sin vow. B,{1- (=) | , 113
ra) {
: [==] v 2m
. us —
log, sin v=log, v-S,, (=) Byes sme 114
Tv
< au = l re n—-1i 2m ~
log, tan xv=log, v+S,, (=) 29, (-1" 2 115
T m
,2m — 1 vn
tan = S,(-1)"" M (=1y- :
“Qm-1
co
If «<1, tan-'!2#=§,(-1)"”’. 2m—1-
INDEX. 185
ART. PAGE
T © GO4)F a a2 1
300. —=,8.,,(—1)”~'.———-ss,, (-1)”"?. — + :
oe Su(=1) 2m—1 Su (1) (239)°""!(2m-—1)
116
300,1. log” = 2nlog 4+4log |n—log |2n—log (2n+1), (w= )-
157
Im— N=
301. & "41<P, (@=— 8). 117 & 158
“6 2m—1 — 2m—1
302. “"+1=P,, }#—(cos Fr ee ae 1 jS100 Sige
n
in 2m—1
303. If n is even, 2°+1=P,,(#?-22x.cos a+l1);
+(n—1) ee
if n is odd, #"+1=(#+1).P,, (#22. Re tl).
n UE A
304. 2”-1=P,,(«—-e ” ye 118 & 158
n —
305. a" —1=P,,$x- (ee eu aR ae 1).
tn—1
306. If is even, v’—1=(#-1).P,, (2 Soa cese i +1);
2 (n—1)
if n is odd, #—1=(#-1).P,,(a@’-22. dee
Tr
+1).
2: r 9 m—1.2 0
306,1. x" —2c0s0.a"+1=P,,(«?—24.cos ———-~ +1). 158
n
cee
307. (14¢.c082)"=1+ Sy} tee. (Le)
(my
a n
+2cosmx.S, m+2r—2 Geyer 119
|m+r—1. r
* Machin’s Theorem.
AA
186 INDEX.
ART. PAGE
308. If y=z+.siny, where x independent of x, then y=
G 2 1 g |2m—1
mes S,(-1)""). ——.. (2 = 1)" -*. sin (2r —1)
|m—r
42 ae 1) ae
| 2mm
ly 2m
™m ne
2 i ges ean ia 1) ) ee (2r)?"— sin@rz. 120
‘he S. sa | 2m oY) mr Oe
309. In the same case, cos y=cosz—w.(sinz)’
2m+1
ues 1)"- } m—n+l .(2n—1)*"~1. cos(2n—1)#
_|m—n+1
Ec ues m+1 [2 m+2
Subs Dae ete SED () wos2a ee
| 2m m—n-+ 1
310. lz =log.(v+f/ v*+1). 123
i)
eee 1)”. [2m
311 = l ae Lal.
: ——— =log, —:
oar lea” x l4+/f lta”
1 ax
312. fr =log, tan — 124.
7 sin & 2°
1 qr aD
313. Hk =log,.cot |— ——}.
7 COS & A 42
|n-1 n—1
x” S = = be i gn —2r
314 fs =— 1—a meres Qe ES = a =
/ 1-2 "|n [a Yen/1—a*
m,—2 r,-2
INDEX. 187
ART. PAGE
n—l |
v zn = : i,
8315. if ae ees BN A — a. jen ie in, 2 F sin-!a, or
2 | | Q
M,—2 $n,2
or
2 1 m—1,—2 -2
=—4/1_-a.S ph om aa 12
316. EE =e = vr S m—1,—2 1 fe
r J 1 ie i SAIL nil 7 oem +t
m,—2
i ——— 126
|2— 1 a Cana 1 — 2
n |n—2
A a in — 1
Sileie — Se pe Mia 3
7 i : 1-—<.S,— erate
a 1l-—wav nm—-1 w@
m,—2
|»—2 1
#(n—1) a 1 LE v
2 m—1,—Z —1);2
i 1-2’ .S,, n =S—5 +e log. Dan hie age WAT
m,—2 £ (n—1), 2
|m—1
Le —_———___
318. f.(sina)” =—Co0s?. 5), oa (sins 2"
[Ss
m,—2
|~—1
eo ‘fGen.
n
r,—2
n—1 1
en l oe
319. f,(sinv)"=—-cosa. are Guiayectt ss Ber Por
n [2
m,—2 $n,2
n—1
t seen
= aaa sy (smae)y?ae™ +t, 128
m,—2
188 INDEX.
ART. PAGE
n—2
; a! Ee J
ro) 3 een oem |
S34 0). f,(sin v) =—Ccos@. eo (sin 1 areca
m,—2
[2-2
4 fsine —(n—2r) 128
n—1 J )
1,—2
|n—2
in
321. i (sina)~"=—cosa.§S,, ue
il
ote (sinanytn te fn
m,—2
n—2 1
+ (n—1) —> 1 L aS v
2—1,—2 + (n—1),2
=-—COS7 1. + <= log,. tan = ahe 129
Ral) (Sina )ee 2 2
m,—2 +(n—1),2
m—1
2 “ —1,—2 n—2m+1
322. f,(cosv)"=sine.S,, =. (cosa)"~*
|
m,—2
n—1
r,—2
lm—1 1
tn [ ie ca
€ 7 1,2 mn m\n—2 92 A -
323. f. (cos #)"=sin Hore oe (cos ny yetet 0, ol
n 2
m,—2 tn,2
n—1
+ (nt)
. — aly —S,.”)
=sin v.§,,7——. (cos x)" *"*). 130
| 2
m,—2
n—2
324. f,(cos 2) -"=sina. SS cooled
1 (cos me 2m + (a LD ee
m,—2
n—2
ace . f, (cos Wa rela:
Ding —2
INDEX. 189
d
ART. PAGE
|[m—2
$n ——. 1
205 a VL ee X m—l,—2 :
325 (cosv)-"=sinw.S,—_— - >? 9!
EC ) 7 n—1 (cosa ne
m,—2
|—-2 ie
(m—1) — I
: ~ 5S
=sINv. So
396. {(Gmay=(21)".25"*". S,.(-))" | ——
Oyo
20%
Seine
=( Sr) oe Seth m—l
mis Cn
+ 42 log ocot (= = ;) - 131
oO 4
9)
7. ~
wi (cosa) *
4 (n—1),2
|
in
sin (7—2m+2)@
|m—1 n—-2m+2
|
+27": ie @, OY
on
\m
3 (n41) bi
cos(w—-2M+4+2)v
im—1 m—2MmM+2
n ,
1S (= sin(n—2m+2)e
(cos x#)"=2°"* Peli tp a NS cE
E( wv) Ol Peels
|
PEG Te or
tn
2%
WD
3 (n+1) | sin (n-2m+2) x
—g—nt+) Ss m—1 132
=) eo
iM |m—1 n—-2M+2
a (tan FA) maa 1
f.(taney'=Sn(-1)"™. (= Gumi
159
n—-2mM+1
(tan a)” —2m +1
m—2mM+1
in
[Ganey Sx Gaye:
(= 0) "a, Or
4(n—-1)
SS. (2n)" at.
(tan iD mat +1
4(=1)'"" log.cosa;
n—-2m+1
Bnei
+(—1)".j,(cot #)’~*.
[.(cotay=Bq (1)
—2m-+1
328.
329.
334.
INDEX.
PAGE
: ae (covaytr"?? r
(COtd)"= 5, et ad OG
J.(cotay'=Su(=1)" 5 (i).
+ (n—1) (cot 2 aie ;
=m ( 1)”. Lana +(-1)?”"” log, . sin a.
160
[Ke Wsa7 SoCs ae (logue), du
+(-1)’. (log, a)’. [,(a".d," 2). 132
Ny n+l
j(a*.«")=a".S,,(-1)""".|n nae ae CO mm) For
m—1
a v.log.a)”
{= =log, eee (eee 133
2 mM. | m
[@.a)=a.S,C ty. dogay ju
+(-1)’. (log, a)’ f(a”. [" 2).
a” ee met ae = ]
=—a". m lo a Ee ag ft.
i Sn (log. a) [n=l
1 = OYA) Ve
log, cee 22 y" 134
, log, m|m
flog, (1 +e cos #)=—a. log, }2e71 (e7!— /e-*=1)}
sin Mv
28, (-1)"7!. (et E71" : ‘
V BE
G, (4,+0,) = Gy. U,+ Gy. ¥,. 136
G,(au,)=a.G,u,; a being independent of ¢ and .
If G,.u,=G,.v,, then shall w,=v,. 137
O. Gy Mg= Gy Un
(¢-'=1)". G;:4,=G,. A,?.U,.
g” (¢-'-1)". Gy.u,=G,. AU, _,-
ART.
343.
344.
345.
346.
SAT.
349.
INDEX. 191
n+l
ae An —-1 Ql
= = €
( See fun} | ‘s G; Uz= G; 4 nm Un —1° Urt+m—-1" 138
n+l
TE eS a bee 1s aD. We, — tee tnien
will ‘
n+l a r
a m—1 r
(S. Fai | Gy. t= Gis Was
=} fa rate i ° Pp q
(¢ =1)P Dna p"- t G,.U,=G,.O. Vis Urs
n+1 ne
=S.eiy*. ee *Usin—m+1°
| m— 1
n+1 |
Ql i —
Us4n=Om —— et "Uy 139
m—1
|w+mr—1
ean ee Py eae Delle. any
m
integer.
Uy ,=(NF1I) 8 IDES wl pevs aan ag
st Nee Ue soa 140
2m —2 1 2n,
C AN i Uy—m+1 st oOs Uy—n
= 2m — «
) Ne NL emery ct Ny iy Tia ° 145
361.
362.
INDEX.
PAGE
m m—r+h
es. WV, atest Orepa10mr PL Oe, where
Vt, = Sy ee aN Bete ee ea
G,(u;).G,(;)=G,. G,(u,.0,)- 146
So" bP GU) hag O,) = G; Grieg: Oo)
(s~'—1)".(¢-1-1)”. G,(4,) . G,(v,)
=G,.G, , A" Gi) Aor
(Sie t 1_1)".G,(u,).G,(v,)=G,. G,A,”. (Uz02)-
n+] =
Az" (4,02) = Spa Nee Nye tly N tae, 147
re ea Gy,4- Us, y= eg aay sane
(cays ; Cai Go Ug Ge : Au
(Get a NG Ay 2. y= Gur,
ry" “2, y°
i <4 Le 2
p—1 = ay =
Unt m,ytn= SA P aN 4 ‘4
rae ren : oor Pas. 148
'S 2
n m—) m—1
ayy 1 mn (— 1) om «Unt n—m4tl,ytn—m+l
| m— 1
Lately published, by the same Author,
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A Shortt GRAMMATICAL INDEX to the Hebrew Text of
the BOOK OF GENESIS; to which is prefixed a Compendious
HEBREW GRAMMAR: designed to facilitate the progress of
those who are commencing the study of the Hebrew Language.
Preparing for the Press,
A TREATISE ON ALGEBRA.
Being a complete Course of Abstract Analysis.
This Work will comprise, in addition to the ordinary Algebra,
the Calculus of Finite Differences, the Differential and Integral
Calculus, and the Calculus of Variations.
Wak aid Lee
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