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DIFFERENTIAL CALCULUS
FOR
BEGINNERS.

DIFFERENTIAL CALCULUS
FOR
e i “ e
* : * * *
BEGINNERS.
BY Aer ...’
< 5 A
º
JOSEPH EDWARDS, M.A.
$ºg
FORMERLY FELLOW OF SIDNEY SUSSEX COLLEGE, CAMBRIDGE.
340mlſon :
MAC MILLAN AND CO.
AND NEW YORK.
I893
[All Rights reserved.]
(Tambridge:
PRINTED BY C. J. CLAY, M.A. AND SONS
AT THE UNIVERSITY PRESS.
*… * 3
º
*
ſy % 03-2
/o - / 3 - // 2.3
PREFACE.
THE present small volume is intended to form a
sound introduction to a study of the Differential Cal-
culus suitable for the beginner. It does not therefore
aim at completeness, but rather at the omission of all
portions which are usually considered best left for a
later reading. At the same time it has been con-
structed to include those parts of the subject pre-
scribed in Schedule I. of the Regulations for the
Mathematical Tripos Examination for the reading of
students for Mathematical Honours in the University
of Cambridge.
Particular attention has been given to the examples
which are freely interspersed throughout the text. For
the most part they are of the simplest kind, requiring
but little analytical skill. Yet it is hoped they will
prove sufficient to give practice in the processes they
are intended to illustrate.
It is assumed that in commencing to work at the
Differential Calculus the student possesses a fair know-
vi PREFACE.
ledge of Algebra as far as the Exponential and Logarith-
mic Theorems; of Trigonometry as far as Demoivre's
Theorem, and of the rudiments of Cartesian Geometry
as far as the equations of the several Conic Sections in
their simplest forms.
Being to some extent an abbreviation of my larger
Treatise my acknowledgments are due to the same
authorities as there mentioned. My thanks are also
due to several friends for useful suggestions with regard
to the desirable scope of the book.
Any suggestions for its improvement or for its
better adaptation to the requirements of junior
students, or lists of errata, will be gratefully received.
JOSEPH EDWARDS.
80, CAMBRIDGE GARDENs,
NoFTH KENSINGTON, W.
December, 1892.
CONTENTS.
CHAPTER I.
LIMITING WALUES. ELEMENTARY UNDETERMINED FORMs.
PAGE
Definitions e iº g & e ë e & g 1—5
Four important Limits . º & & º 6—7
Undetermined forms: method of procedure . & & 8–11
CHAPTER II.
DIFFERENTIATION FROM THE DEFINITION,
Tangent to a Curve. Definition. Equation . º e 13–14
Differential Coefficient . § tº e e & * 14—17
A Rate-measurer & & s º e g & e 18–20
CHAPTER III,
FUNDAMENTAL PROPOSITIONs,
Constant. Sum. Product. Quotient . t * * 22—25
Function of a function . * g * g * # 27—29
viii coyTENTS.
CHAPTER IV.
STANDARD FORMS,
Algebraic, Exponential, Logarithmic Forms
Direct and Inverse Trigonometric Functions .
Table of Results
Logarithmic Differentiation
Partial Differentiation
CHAPTER W.
SUCCESSIVE DIFFERENTIATION.
Standard results and processes
Use of Partial Fractions and Demoivre's Theorem.
Leibnitz's Theorem
Note on Partial Fractions
CHAPTER WI.
ExPANSIONS.
Elementary Methods.
Taylor's Theorem
Maclaurin’s Theorem • tº º e g
By the Formation of a Differential Equation.
Differentiation or Integration of a Known Series
CHAPTER VII.
INFINITESIMALs.
Orders of Smallness. Infinitesimals
Difference of Small Arc from Chord
PAGE
31–33
34–38
39–40
40–41
44–47
52–54
55–56
57–60
61–63
66–68
69–70
71–73
73–77
77–79
83–86
87
COWTENTS.
ix
CHAPTER VIII.
TANGENTS AND NORMALS.
K h
Equations of Tangent and Normal .
Cartesian Subtangent, Subnormal, etc.
Values of # da: etc.
a ’ dº ’
Polar Co-ordinates, Subtangent, Subnormal, Angle between
Radius Vector and Tangent, etc. -
Pedal Equation . e • . & e e te ©
Maximum number of Tangents or Normals from a given
point .
CHAPTER IX.
ASYMPTOTEs.
Methods of finding Asymptotes of a Curve in Cartesians.
Polar Asymptotes
CHAPTER X.
CURVATURE.
Expressions for the Radius of Curvature.
Co-ordinates of Centre of Curvature
Contact
CHAPTER XI.
ENVELOPES.
Method of finding an Envelope
PAGE
90–96
97–99
100—101
102–105
106–107
... 108–110
115—125
126–129
132–150
152–153
153–158
CHAPTER XII.
Associatel) LocI.
Pedal Curves
Inversion .
Polar Reciprocal
Involutes and Evolutes
162–168
173–179
180–181
182
183–187
X COWTENT'S.
CHAPTER XIII.
MAXIMA AND MINIMA.
PAGE
Elementary, Algebraical and Geometrical Methods . . 192–196
The General Problem g e g e º fº , 197—206
CHAPTER XIV.
UNDETERMINED ForMs.
Applications of the Differential Calculus to the several
forms . * * ſº º © & * {e , 210–219
CHAPTER XV.
LIMITATIONs of TAYLOR's THEOREM.
Continuity. e & e 222–224
Lagrange-formula for Remainder after m terms of Taylor's
Series. • . . tº { } * t & º , 225–230
*
* \
\º
DIFFERENTIAL CALCULUS,
CHAPTER I.
LIMITING VALUES, ELEMENTARY UNDETERMINED
FORMS.
1. Object of the Differential Calculus. When
an increasing or decreasing quantity is made the subject
of mathematical treatment, it often becomes necessary
to estimate its rate of growth. It is our principal object
to describe the method to be employed and to exhibit
applications of the processes described. -
2. Explanation of Terms. The frequently re-
curring terms “Constant,” “Variable,” “Function,” will
be understood from the following example:
Let the student imagine a triangle of which two sides a, y are
unknown but of which the angle (A) included between those sides is
known. The area (A) is expressed by -
A=#vy sin A.
The quantity A is a “constant” for by hypothesis it retains the same
value, though the sides a, and y may change in length while the
triangle is under observation. The quantities a, y and A are there-
fore called variables. A, whose value depends upon those of a; and y,
is called the dependent variable; a. and y, whose values may be any
whatever, and may either or both take up any values which may be
assigned to them, are called independent variables. - -
The quantity A whose value thus depends upon those of ac, y and
A is said to be a fumction of a, y and A. r
E. D. C. Q: l
2 - DIFFERENTIAL CA LOUI, U.S.
3. Definitions. We are thus led to the following
definitions:
(a) A CONSTANT is a quantity which, during any set
of mathematical operations, retains the same value.
(b) A WARIABLE is a quantity which, during any set of
mathematical operations, does not retain the same value
but is capable of assuming different values.
(c) An INDEPENDENT WARIABLE is one which may
take up any arbitrary value that may be assigned to it.
(d) A DEPENDENT WARIABLE is one which assumes
its value in consequence of some second variable or system
of variables taking up any set of arbitrary values that
may be assigned to them.
(e) When one quantity depends upon another or upon
a system of others in such a manner as to assume a
definite value when a system of definite values is given to
the others it is called a FUNCTION of those others.
4. Notation. The usual notation to express that
one variable y is a function of another w is
y = f(a) or y = F(a) or y = b (a).
Occasionally the brackets are dispensed with when no
confusion can thereby arise. Thus fu may be sometimes
written for f(a). If u be an unknown function of
Several variables a, y, z, we may express the fact by the
equation u = f(a), y, z).
5. It has become conventional to use the letters
a, b, c... a, S, ſy... from the beginning of the alphabet
to denote constants and to retain later letters, such as
ºw, v, w, aſ, y, z and the Greek letters É, m, & for variables,
6. Limiting Values. The following illustrations
will explain the meaning of the term “LIMITING
WALUE * :
LIMITIAWG VA.J., U.E.S. 3
(1) We say '6=3, by which we mean that by taking enough sixes
we can make '666... differ by as little as we please from #.
20; -- 3
a; + 1
For the difference, between 2a: + 3
(2) The limit of when a is indefinitely diminished is 3.
# and 3 is #1, and by diminishing
a; indefinitely this difference can be made less than any assignable
quantity however Small.
3
2 + -
The expression can also be written —#, which shews that if a.
1+.
be increased indefinitely it can be made to continually approach and
to differ by less than any assignable quantity from 2, which is there-
fore its limit in that case.
It is useful to adopt the notation Lt.-a to denote
the words “the Limit when a = a, of.”
2 ſº
Thus Lt.-0 ***=3; Lt.-- *#3_2.
a; + 1 a; +1
(3) If an equilateral polygon be inscribed in any closed curve and
the sides of the polygon be decreased indefinitely, and at the same
time their number be increased indefinitely, the polygon continually
approximates to the form of the curve, and ultimately differs from it
in area by less than any assignable magnitude, and the curve is said to
be the limit of the polygon inscribed in it.
7. We thus arrive at the following general defini-
tion:
DEF. The LIMIT of a function for an assigned value
of the independent variable is that value from which the
function may be made to differ by less than any assign-
able quantity however Small by making the independent
variable approach sufficiently near its assigned value.
8. Undetermined forms. When a function in-
volves the independent variable in such a manner that
for a certain assigned value of that variable its value
cannot be found by simply substituting that value of the
variable, the function is said to take an undetermined
form.
1—2
4. DIFFERENTIAL CALC'ULUS,
One of the commonest cases occurring is that of a
fraction whose numerator and denominator both vanish
for the value of the variable referred to.
Let the student imagine a triangle whose sides are
made of a material capable of shrinking indefinitely till
they are smaller than any conceivable quantity. To fix
the ideas suppose it to be originally a triangle whose
sides are 3, 4 and 5 inches long, and suppose that the
shrinkage is uniform. As the shrinkage proceeds the
sides retain the same mutual ratio and may at any
instant be written 3m, 4m, 5m and the angles remain
unaltered. It thus appears that though each of these
sides is ultimately immeasurably small, and to all prac-
tical purposes zero, they still retain the same mutual ratio
3 : 4: 5 which they had before the shrinkage began.
These considerations should convince the student
that the ultimate ratio of two vanishing quantities is not
necessarily 2ero or unity.
o
4.4
e . aft — a is gº
9. Consider the fraction ; what is its value
when a = a 2 Both numerator and denominator vanish
when a is put = a. But it would be incorrect to assume
that the fraction therefore takes the value unity. It is
equally incorrect to suppose the value to be zero for the
reason that its numerator is evanescent ; or that it is
infinite since its denominator is evanescent, as the
beginner is often fallaciously led to believe. If we wish
to evaluate this expression we must never put aſ actually
equal to a. We may however put a = a + h where h is
anything other than Zero. *
Thus
a;” — dº
= 2a + h,
Q} – O.
and it is now apparent that by making h indefinitely
small (so that the value of a is made to approach inde-
finitely closely to its assigned value a) we may make the
expression differ from 24 by less than any assignable
EXAMPLES. 5

qu tity. Therefore 2a is the limiting value of the
fwen fraction. *
10. Two functions of the same independent variable
are said to be ultimately equal when as the independent
variable approaches indefinitely near its assigned value
the limit of their ratio is unity. c
Thus Lto-0 º = 1 by trigonometry,
and therefore when an angle is indefinitely diminished
its sine and its circular measure are ultimately equal.
EXAMPLES.
1. Find the limit, when a = 0 of #.
(a) when y=mw,
(b) when y=a^/a,
(c) when y=a^* +b.
. . . . aw-H b : .. * &
2. Find J.t #. * (i) when waO, (ii) when v = ob.
& w8 – a 3 - a;4 – a 4 a;6 — aft
3.--Find Lt.-a-º-; Lt.-aº - ; Lt, a 7.2. T. 2 .
gº." Jº — C, 30 — C, 3/4 — C,
4, Find the limit of c. * (i) when a - 0, (ii) when a = ob.
Cà) -- —
3}
-N s 3a*— 4a: + 1
5. E ind Lº-1 a;3–4a, E3 *
6. The opposite angles of a cyclic quadrilateral are supple-
mentary. What does this proposition become in the limit when
two angular points coincide?
a;3–63:?-- 11a; - 6
7. Evaluate the fraction ac-3 – 642-2+ 11a2-1–
6 for the values
w-co, 3, 2, 1, #, #, 0, -oo.
8. Evaluate Lt. # and 1.
WT-Fºr-1
* Win || 1 a
6 J)IFFERENTIAL CALOUI, U.S.
11. Four Important Limits. The following
limits are important :
(I) L.” = 1 ; Lto-ocos 6 = 1,
(II) Lt.-1 a – 1 = ??,
a;” — I
Q;
(III) Ltº-2 (1+. ) = e, where e is the base of the
{!} Napierian logarithms,
a" —
(IV) Lt...". -logo.
12. (I) The limits (I) can be found in any standard
text-book on Plane Trigonometry.
13 (II) Toprowell-º-;
when aſ approaches unity 2 approaches zero. Hence we
can consider 2 to be less than 1, and we may therefore
apply the Binomial to the expansion of (1 + 2)” what-
ever n may be.
=n. Let al-1+2. Then
a!" – 1 . . (1 + 2)” – 1
Thus L., E.-L. "+"
n: "*****.
= Lt.-0 +
= Lºt".”...}=n.
14. (III) To prove 1...( + !) = 6.
2:
I
Let y=(1+. 3
then loge y = w logo (1 + ...) *
FO UAE - IMPORTAMT LIMITS, 7
Now w is about to become infinitely large, and there-
| - + §
fore may be throughout regarded as less than unity,
and we may expand by the Logarithmic Theorem.
I I I
Thus ºv-º-,+,+- º
{0
- I 1.
~ *- : +3,. - …
I e -
= 1 – 1. ^ [a convergent Series].
Thus when w becomes infinitely large
Lt logay = 1,
and Lt y = e,
i.e. Lie-w ( + y F 6.
Q}
Q} ) (!,
COR. 1.--(1 + ..) = Lt. (l –H “) = 6%.
- £ :=% &\}
-- - a" — I .
15. (IV) To prove Lt.-0 →-- logea,
Assume the expansion for aº, viz.
a;2
a”=1+ a loged + 2. (logea)* + ...,
which is shewn in Algebra to be a convergent series.
a” – 1 log, a)”
Hence ===logo **** + ...
= loged +a; x [a convergent Series].
And the limit of the right-hand side, when a is in-
definitely diminished, is clearly log.a.
8 DIFFERENTIAJ, CA. LCULUS,
16. Method of procedure. The rule for evaluating
* * ge tº ()
a function which takes the undetermined form when
() -
the independent variable a ultimately coincides with its
assigned value a is as follows:–
Put a = a + h and eaſpand both numerator and demo-
minator of the fraction. It will now become apparent
that the reason why both numerator and denominator
ultimately vanish is that some power of h is a common
factor of each. This should now be divided out. Finally
let h diminish indefinitely so that a becomes ultimately
a, and the true limiting value of the function will be
clear. -
In the particular case in which w is to become 2ero
the expansion of numerator and denominator in powers
of a should be at once proceeded with without any pre-
liminary substitution for w.
In the case in which a is to become infinite, put
I
a = −,
$/
so that when a becomes o, y becomes 0.
Several other undetermined forms occur, viz. 0 × 00,
. oo – co, 0", oo", 1", but they may be made to depend
upon the form . by special artifices.
The method thus indicated will be best understood
by examining the mode of solution of the following
examples:—
- * a;7 – 2a:54-1
Tºx. 1. - Find *-iji-ji; .
This is of the form . if we put a = 1. Therefore we put a = 1 + h
and expand. We thus obtain -
I,t *-*#4 El (1+h)7-2 (1+h)*4-1
.1 1 03; - 1 a;3 --- h = 0 (1+ h)” – 3 (1+h)*4-2
– 3a.” + 2
UAWDETERMINED FORMS. 9
= Lt (1+7h +21h” +...) – 2 (1 + 5h + 10h.” +...) + 1
-*-*TúI5. I5RT.)=5 (IIERIN).I.
–3h--h”--...
= Lt.-0 – 3h + - -
~ 3 + h_--
=Lt.-, -āj-
— 3
= — = 1
– 3
It will be seen from this example that in the process of expansion
it is only necessary in general to retain a few of the lowest powers of h.
30 — h9%
Ex. 2. Find Lt.-.” &
Here numerator and denominator both vanish if a be put equal to 0.
We therefore expand a' and b* by the exponential theorem. Hence
- - 30 7,90
Lt.-, * *
3, .

ſ
2.2 .2 -
º logea +: (logaa)*-H ...] -- | +a; logob +. (logob)*-- º
Q.
= Lt.-0
= Lt.-0 |owa – loge b + s (log, al” –log.blº) -H . . }
(I, - -
=logea – logeb-logo b'
I
* /tan a Nº.
Ex. 3. Find Lt.-0 ( {C g
& - - tan aſ 1 sin a
Since & . = −. . --— , ,
Q} COS (E (C
tana: -
we have Lt.-0 = 1.
|
tan as
Hence the form assumed by (
) is an undetermined form 1°
when we put a = 0.
Expand sin a and cosa in powers of w. This gives
l £ £1. \;
--- - -, -ī- . . . W3C
Lt tan as "—Lt - 3!
a: = 0 Q} — Li va: == 0 a;3
*-gi-F.
I () DIFFERENTIAL CA/CULUS,
I
.2 22
= Lt.-0 (1+; + higher powers of •)
1
a;3 2
=Lt.-0 | tº (1+ …)
1
27\ z2
—Lt.--(1+...) y
where l is a series in ascending powers of a whose first term (and
therefore whose limit when a -0) is unity. Hence
1 3 t
22 a;2|\iº 3
1...(**) = Lt.-0 (1+...) | =es, by Art. 14.
&
l
Ex. 4. Find Lt.-1a:F.
This expression is of the undetermined form 1°.
Put 1 — a = y,
and therefore, if a = 1, y =0;
l
therefore Limit required=It, o (1 – y)= e−1 (Art. 14).
I
Ex. 5. Lt.-...a (a^-1).
This is of the undetermined form ox 0.
Put . 2– 5
$/
therefore, if a. = Go, y=0,
W –
and Limit required=It,-0 (l, 4/ l = logoa (Art. 15).
17. The following Algebraical and Trigonometrical series are
added, as they are wanted for immediate use. They should be learnt
thoroughly.
n (n-1)...g., n (n − 1)(n-2)...a
-j-a-º-F-I-5 a-w' ......
-, + , ,,..., n (n+1), a n (n+1) (n+2).3.
(1 – a.) "=1+ng tº QC +–Hºa– a + . . . . . .
až (logea)* , aº (logea)*
(1+a;)”=1 + ma;+
2] 3! ' ' ' ' ' '
a;2 agº
*=1+z-Hi. + 3 + tº e º e º 'º'
EXAMPLE'S. j 1
l 36. 1 - -- - -- - - -T- . . . . . .
oge (1 + æ)=a. 2 + 3 - 4 +
a;2 3.3 a;4
log, (1 – a = – a 2 T 3 T 4. T ''''''
2 ge I.-w 3 5 e - A & 4 &
a;3 agö
~1 or — or — - -
tan-ºw-º-; +H -......
.2 a;4
cost-1-3. t IIT * * * * * *
- a;3 agº
sm *-*-āli Hi- • a tº “ - 4
.. cº-l-e-º a;2 agł
w E–a– l = 1 + - Tº Tú" , , , , , .
cosh a [whic 2 | *2] + 4! +
30 — a 3. 3 or 5
inhºſ which=ºl-zº
EXAMPLES.
Find the values of the following limits :
1. *- := 2 4.- := 3. *- :-}
I
4. L...."tºº. 5. L.-èº
6. L.-è *::::: ** 7. L. ºº
8. L.-,++*. 9. a. *-*.
10. a...ºr-lºgº. 11. 1...º-infº.
12. L. "...” 13. L. “º
14 a.º. is 1...ºne
e * \ 0=z
2. (. a) AT
I
ºr Dºo-ººr st
ga' – a – a UIS 22
'9% '*(a stoaoo)"""T
T
ë ... ( & ) 0 =::
†Z º UIS, 2T
T
e e Q? 0 = 0:
ZZ, º a) 17
T
g tº 13' 0 = 0:
$2:
g? 0=3:
!
w8–ºoté+, usz T
a!-H I I e
gº 0 = *AT
É. f_UIS – (a +I)”3ol-agsooar
gº
'S'ſ) Tſ)0TWO TWILIAWTºſſºſ'ſ IOI
“GZ
‘92,
“Tº
'6T
‘9T
&I
CHAPTER II.
DIFFERENTIATION FROM THE DEFINITION.
18. Tangent of a Curve; Definition; Direction.
Let AB be an are of a curve traced in the plane of the
paper, OX a fixed straight line in the same plane. Let
Y B
wº
, ºr 1.
&
O | T M N X .

P, Q, be two points on the curve; PM, QN, perpen-
diculars on OX, and PR the perpendicular from P on
Q.N. Join P, Q, and let QP be produced to cut OX
at T. - • * >
When Q, travelling along the curve, approaches in-
definitely near to P, the limiting position of chord QP is
called the TANGENT at P. QR and PR both ultimately
vanish, but the limit of their ratio is in general finite;
for Lift = Lt tan RPQ = Lt tan XTP = tangent of the
angle which the tangent at P to the curve makes with
OX.
14 DIFFERENTIAL CALCULUS,
If y = p(a) be the equation of the curve and a,
a + h_the abscissae of the points P, Q respectively;
then MP = p(a), NQ = q (a + h), RQ = q.(a + h) — (p(a)
l.
and PR =
Thus Lº-Lº-º-º:
sm-->
PR
Hence, to draw the tangent at any point (a, y) on the
curve y = p(a), we must draw a line through that point,
making with the axis of a an angle whose tangent is
a perº-ºº.
and if this limit be called m, the equation of the tangent
at P(a, y) will be
Y— y = m (X. — a.),
X, Y being the current co-ordinates of any point on the
tangent; for the line represented by this equation goes
through the point (a, y), and makes with the axis of a
an angle whose tangent is m.
19. DEF—DIFFERENTIAL CoEFFICIENT.
Let p(a) denote any function of w, and b (a + h) the
Same function of a + h; then Lth-0 q (a + º — q (a) is
called the FIRST DERIVED FUNCTION or DIFFERENTIAL
COEFFICIENT of 4 (a) with respect to w.
The operation of finding this limit is called differ-
entiating ºp (a). .
20. Geometrical meaning. The geometrical
meaning of the above limit is indicated in the last article,
where it is shewn to be the tangent of the angle \, which
the tangent at any definite point (a, y) on the curve
y = p(a) makes with the aa is of w.
21. We can now find the differential coefficient of
any proposed function by investigating the value of the
GEOMETRICAL MEANING. 15
above limit; but it will be seen later on that, by means
of certain rules to be established in Chap. III. and a
knowledge of the differential coefficients of certain
standard forms to be investigated in Chap. IV., we can
always avoid the labour of an ab initio evaluation.
}
Ex. 1. Find from the definition the differential coefficient of .
where a is constant ; and the equation of the tangent to the curve
ay=a.”. -
a;2
* , * @)=;
ſº ‘p (**)=" fºr 3.
- Arh iſ 67* 2 — or 2
therefore I,th-o b (v-El) – b (a) = Lt.-0 (a + h)*-wº
h ha
2a:h + h^ 2a: + h)
=Lt.-0 ha - -" h)
_2~
T a .
The geometrical interpretation of this result is that, if a tangent
be drawn to the parabola ay=a.” at the point (a!, y), it will be inclined
to the axis of a; at the angle tanT" * [.
The equation of the tangent is therefore
2a:
Y-y== (X– ar).
Ex. 2. Find from the definition the differential coefficient of
iº 4: tº
logº sing , where a is a constant.
Here ºb (a)=loge sin . 3.
and
log, sin 4-F h_ loge sin a.
ºp (a + h) – ºp (a) (l (l,
Lt.-0 - h + = Lt.-0
h
. (C h a: . h.
sing cos; + cos; sing
=Lt.-0; log.
. 30
SIIl –
(I,
= Lt.-0 } loge ( +. Cot — higher powers of h)
I6 DIFFERENTIAL CALOULUS,
|by substituting for sin . and cos . their expansions in powers of |
} ge
“cot . – higher powers of h
(l,
h
[by expanding the logarithm]
1 Q}
= - Cot, - .
0. (l,
= Lt.-0
Hence the tangent at any point on the curve #=log, sin; is inclined
to the axis of a; at an angle whose tangent is cot ;: that is at an angle
: --- . and the equation of the tangent at the point a, y is
. Y-y=cot: (X-a;).
EXAMPLES.
Find the equation of the tangent at the point (a, y) on each
of the following curves: -
1. )=w8. 2. y=a^. 3. */=War.
4. W=a^+a;”. 5. $/=sin w. 6. /=e".
7. /=log, w. 8. Aſ =tana. 9, a 2+y}=6%.
10. wº/a2+y^{b}=1.
22. Notation. It is convenient to use the notation
ða for the same quantity which we have denoted by h,
viz. a small but finite increase in the value of a. We
may similarly denote by 8y the consequent change in the
value of y. Thus if (a, y), (a + 6a, y + 8/) be contiguous
points upon a given curve y = q (a), we have
- y+ 8y = {(a + 8a),
and by = p (a +8a)– b (a).
Thus the differential coefficient
Ltº-0 q (a + º – b (a)
NOTATION. 17
may be written
öy
8a,
which more directly indicates the geometrical meaning
RQ
PR
Ltº-0
Lipº-0
pointed out in Art. 18.
The result of the operation expressed by
1...º-ºº.
or by lºw-0 º 2
e d dy
is denoted by da, " " do
The student must guard against the fallacious notion
that da; and dy are separate small quantities, as 8a, and
ðy are. He must remember that dw is a symbol of
operation which when applied to any function (b (a)
means that we are \\
(1) to increase w to w ł. h.
(2) to subtract the original value of the function,
(3) to divide the remainder by h94
(4) to evaluate the limit when h ultimately vanishes.
Other notations expressing the same thing are
a;) d f : A e
º ), #. ºp'(a), $', 'b, ‘ba, $/, /, /1.
EXAMPLES.
...a dº/ . e -
Find i. in the following cases:
1. y=20. 2, 3/= 2+2. 3. 3/=2+3a.
E. D. C. 2
, , ,
; : " ' |
18 DIFFERENTIAL CA LOUI, U.S.
I }
4 y=2+33*. 5. 9–3. 6. v=x-a.
7. v=ºta. 8. y=a Wa. 9. y = Wºº-Faż.
10, ºſ- eV. 11. y =e”. 12. Aſ =log, Sec w.
sin &
13. 3/=a sin w. 14. 4/= 15. 3) =a^.
Jº
23. Aspect of the Differential Coefficient as a
Rate-Measurer. When a particle is in motion in a
given manner the space described is a function of the
time of describing it. We may consider the time as an
independent variable, and the space described in that
time as the dependent variable.
The rate of change of position of the particle is
called its velocity.
If uniform the velocity is measured by the space
described in one second; if variable, the velocity at any
instant is measured by the space which would be
described in one second if, for that second, the velocity
remained unchanged.
Suppose a space S to have been described in time t
with varying velocity, and an additional space 8s to be
described in the additional time St. Let v, and v, be
the greatest and least values of the velocity during
the interval 8t; then the spaces which would have been
described with uniform velocities v, v, in time Št are
viðt and vôt, and are respectively greater and less than
the actual space 88.
S * $
Hence v, , , , and v, are in descending order of mag-
8t 3.
nitude.
If then Öt be diminished indefinitely, we have in the
limit v. = v, = the velocity at the instant considered,
which is therefore represented by Lt. , i.e. by % *
A RATE-MEASURER. 19
24. It appears therefore that we may give anotherin-
ds
dt
the rate of increase of s in point of time. Similarly
% 2 % , mean the rates of change of a, and y respectively
terpretation to a differential coefficient, viz. that iſ means
in point of time, and measure the velocities, resolved
parallel to the axes, of a moving particle whose co-
ordinates at the instant under consideration are a, y.
If a, and y be given functions of t, and therefore the
path of the particle defined, and if 8a, Šy, 8t, be simul-
taneous infinitesimal increments of a, y, t, them
ðy dy
dy r, by , , §t ºf
8; dt
and therefore represents the ratio of the rate of change
of y to that of w. The rate of change of w is arbitrary,
and if we choose it to be unit velocity, then
dy dy
da, T di = absolute rate of change of y.
25. Meaning of Sign of Differential Coefficient.
If a be increasing with t, the a-velocity is positive,
whilst, if a, be decreasing while t increases, that velocity
is negative. Similarly for y.
dy
º dy dt dy
Moreover, since + = + , º is positive when aſ and a
5 d. T J. di. “P !/
dt
ºncrease or decrease together, but negative when one in-
creases as the other decreases.
This is obvious also from the geometrical inter-
* d g g e
pretation of . For, if a. and y are increasing together,
2—2
20 DIFFERENTIAL CALCULUS,
dy
... is the tangent of an acute angle and therefore positive,
da,
dy
while if, as a increases y decreases, da,
tangent of an obtuse angle and is negative.
represents the
26. The above article frequently affords important
information with regard to the sign of a given expression.
For if, for instance, p(a) be a continuous function which
is positive when a = a and when a = b, and if p'(a) be
of one sign for all values of a lying between a and b so
that it is known that b (a) is always increasing or
always decreasing from the one value b (a) to the other
q, (b), it will follow that b (a) must be positive for all
intermediate values of a.
Ex. Tiet q (a)=(a – 1) e”--1.
(a + h – 1) e^+” – (a – 1)e”
h
(a + h-1) (1 + h--...) — (a) – 1) .
Here p (0)=0 and p' (a)= Lth-0
- 1, .ºrhigher powers of h e” – ace”.
h
So that p' (w) is positive for all positive values of a. Therefore as a
increases from 0 to o, ºp (a) is always increasing. Hence since its
initial value is zero the expression is positive for all positive values
of a;.
EXAMPLES.
1. Differentiate the following expressions, and shew that
they are each positive for all positive values of a
(i) (v-2) e”--a-H2,
(ii) (a, -3) *4,420+8,
(iii) a -logo (1 + æ).
2. In the curve y=ces, if / be the angle which the tangent
at any point makes with the axis of w, prove y=c tan W.
Q?
3. In the curve y=c cosh #, prove y=esee W.
Jº VAMPLE'S. 2I
4. In the curve 3b%/=a^–3aw” find the points at which the
tangent is parallel to the axis of w.
[N.B.-This requires that tan V =0.]
5. Find at what points of the ellipse a "ſa”--y”/b%= 1 the
tangent cuts off equal intercepts from the axes.
[N.B.-This requires that tan / = + 1.]
6. Prove that if a particle move so that the space described
is proportional to the square of the time of description, the
velocity will be proportional to the time, and the rate of increase
of the velocity will be constant.
7. Shew that if a particle moves so that the space described
is given by S oc sin put, where p is a constant, the rate of increase
of the velocity is proportional to the distance of the particle
measured along its path from a fixed position.
8. Shew that the function
a sin a + cos & H-cos” &
continually diminishes as a increases from 0 to T/2.
9. If y=2~-tan- w = log,(w-WI-Fº),
shew that y continually increases as a changes from Zero to
positive infinity.
10. A triangle has two of its angular points at (a, 0), (0, b),
and the third (a, y) is moveable along the line y=a. Shew that
if A be its area dA
2 + = a + b,
da:
and interpret this result geometrically.
11. If A be the area of a circle of radius a, shew that the
circumference is #. Interpret this geometrically.
12. O is a given point and WP a given straight line upon
which OM is the perpendicular. The radius OP rotates about
O with the constant angular velocity o. Shew that WP
increases at the rate
a) . ON Secº WOP.
CHAPTER III,
FUNDAMENTAL PROPOSITIONS.
27. IT will often be convenient in proving standard
results to denote by a small letter the function of a
considered, and by the corresponding capital the same
function of a + h, e.g. if u = b (a), then U = p (a + h),
or if u = aº, then U = a "t".
Accordingly we shall have
du U — u
... = Lth-0+--,
da, h
dy V — y
... = Lt.-0–F–
da h
etc.
We now proceed to the consideration of several im-
portant propositions.
28. PROP. I. The Differential Coefficient of any
Constant is zero. This proposition will be obvious
when we refer to the definition of a constant quantity.
A constant is essentially a quantity of which there is
no variation, so that if y = c, Sy = absolute Zero what-
ever may be the value of 8a. Hence º = 0 and
dy = 0 when the limit is taken.
da,
Or geometrically: y = c is the equation of a straight line parallel
to the c-axis. At each point of its length it is its own tangent and
makes an angle whose tangent is Zero with the w-axis.
FUNDAMENTAL PROPOSITIONS. 23
29. PROP.II. Product of Constant and Function.
The differential coefficient of a product of a constant and
a function of w is equal to the product of the constant
and the differential coefficient of the function, or, stated
algebraically,
du,
i;(0)=0 . .
d : c U – cu U – u
For da (Cu) = Lt.-0 — — = clºth-0 -j-
_, du
= C da, &
30. PROP. III. Differential Coefficient of a Sum.
The differential coefficient of the sum of a set of functions
of a is the sum of the differential coefficients of the Several
functions.
Let u, v, w, ..., be the functions of a, and y their
SULTY).
Let U, V, W, ..., Y be what these expressions
severally become when a is changed to a + h.
Then y = u + v + w -- ...
Y = U + V + W -- ...,
and therefore
Y — y = (U – u + (V – v) + (W – wy -- ...;
dividing by h,
Y— y U – u : V – v W — w
# * =– +---|--|--
and taking the limit
dy du du dw
dº do "dº" da
If some of the connecting signs had been – instead
of + a corresponding result would immediately follow,
e.g. if
+ . . . .
y = u + v - w ł.
24 DIFFERENTIAL CALCUI, U.S.
dy du , du dw
31. PROP. IV. The Differential Coefficient of
the product of two functions is
(First Function) × (Diff. Coeff. of Second)
+ (Second Function) × (Diff. Coeff. of First),
or, stated algebraically,
d (w) ... do du
do º "d, tº in
With the same notation as before, let
+ 2)
- r 3) = uv, and therefore Y = UV;
whence Y – g = UV – up
= u (V – v) + V (U – wy;
therefore Y-y = Q(. W – v + V U — w
h h h
and taking the limit
dy_, do , , du
dº “d," "dº
32. On division by uv the above result may be written
1dy 1 dull du
y da u da v da
Hence it is clear that the rule may be extended to pro-
ducts of more functions than two.
For example, if y = www; let ww = 2, then y = u2.
1dy I dull de
y da u da 2 da
I de I do I dw
z da y da; ' nu da
Whence
but
FUWDAMEWTAJ, PROPOSITIONS. 25
whence by substitution
1 dy_1 du , I
y da u da v
du 1 dw
da, "w da
Generally, if $/ = wwwt...
1 dy. I du , 1 du , 1 dw I di
y dº Tu da" a da "w do " i dº "".
and if we multiply by uwwt... we obtain
dy du , , dv dw
da, T (wwt...) do." (wwt...) do." (uvt...) da, " . . . .
i.e. multiply the differential coefficient of each separate
function by the product of all the remaining functions
and add up all the results; the sum will be the differ-
ential coefficient of the product of all the functions.
33. PROP. W. The Differential Coefficient of a
quotient of two functions is
(Diff. Coeff. of Num".) (Demº.) - (Diff. Coeff. of Den”.) (Num".)
Square of Demominator
or, stated algebraically,
d (...) dw * T dº
da \v
Q!,
Q
q}~
With the same notation as before, let
>
%
Qſ,
y = ..., and therefore Y =
Q)
- U aſ
whence Y – aſ = ... —
$/ V v
Uv — Wu
Vy
26 DIFF/ER/EWTIA/, CA LOUI, U.S.
U-u, W = 0,
(*ſy Y-y__l }, “
therefore h. T Vºy j
and taking the limit
du dy
dy de" do."
da, T Q)”
34. To illustrate these rules let the student recall to memory the
differential coefficients of aº and a logo sin; established in Art. 21, viz.
23, and cot a respectively.
Ex. 1. Thus if !/=a.” + a loge sin 7
(I.
dy a
We have by Prop. III. Hº- 2a: + cot . .
da: (l,
Ex. 2. If Q) = wºx a logo sin . 7
dy :.. " . . o . . . .”
we have by Prop. IV. dº 24 x a logesin a + æ” Cot a
aloge sin -
(l,
c. 3. If * I →.
Ex !ſ a;2 3.
a:9. Got a 2a: . a loge sin a.
we have by Prop. V dſ_ (l (l,
- - 1). Y. (lit.: T A:4
EXAMPLES.
[The following differential coefficients obtained as results of
preceding examples may for present purposes be assumed :
— oë – “3 a.2 — oſC — 230
3/= 4", $/l=3a*. 3/=e", 3/1 = €”.
I
3/=a^*, 3/1=44%. 3/=log, w, 91–.
-- l - 9
y=Wa, y= - 8/=tan 4, 3/1=Secº ar.
2 Wº
3/=sin a, 91 = COs ar. /=log, sin w, yi =cot w.]
FUWCTION OF A FUWCTION. 27
Differentiate the following expressions by aid of the foregoing
rules:
a;3 sin ºv, aße", a "logs ar, a" tan ar, a;" logo sin a.
1.
2. a 4/sina, sin ala”, sin a ſe”, e”/sina'.
3. tana. log, sina, 6°log, w, sin” aſcosa.
4. aše” sina, a tan a logo v.
5. a' sin ale", a'ſe" sin a, Iſa’e” sin a.
6, 2A/w.sina, 3 tan w/War, 5+4e"/w/º.
7. e." (cº-i-Wºº), (a8+ æ") (e” logº ar).
35. Function of a function.
Suppose * = f(0) ........................ (I),
where t = h(º)....................... .(2).
If a be changed to a + 8a, v will become v-Höv, and
in consequence u will become u + 8w.
Now if v had been first eliminated between equa-
tions (1) and (2) we should have a result of the form
w = F(a)........................ (3).
This equation will be satisfied by the same simul-
taneous values a + 8a, u + 8w, which satisfy equations
(1) and (2). Also
bu ou 80 .
8a, 8v 8a, ’
and proceeding to the limit
Ltº-0 : - º as obtained from equation (3),
Liao-0 8w _du as obtained from equation (1)
8v dy }
1..." - du as obtained from equation (2).
ôa, da;
du du do
Thus da, ~ dy * da, p
28 DIFFERENTIAJ, CA LOUI, U.S.
36. For instance, the diff. coeff. of aº ; is 2a:,
it. 21.
and of logo sin a is cot ..] Art
Suppose u-(log, sin ay”, i.e. vº where v =log, sina, then
du du dv g
da; T do ’ #=2...dots-2cota.log sing
37. It is obvious that the above result may be
extended. For, if u = p(v), w = |*(w), w = f(a), we have
du du du
da, Tch dºn’
du du du! .
but
da, du da
du du du dw
and therefore ––– = --- . H- . 2
da, dy dw da;
and a similar result holds however many functions there
may be.
The rule may be eaſoressed thus:
d (1st Func.) d (1st Func.) d (2nd Func.), d (Last Func.)
da; T d (2nd Fumc.) d (3rd Fume.) * a da:
or if u = q [A] {F'(fo)}].
du
TV F(ſ)|x º' F(f) < P'(f)x fºr
Thus in the preceding Example
d (log,sina)*_d (log, sin it)” d log, sin a
da: Tº d(log, sin it) da:
= 2 log, sin a . cota.
Again,
d (log, sina.”) d (log, sin wº) d sin a.º. daº
da: T d (sina.”) da:2 da;
= ---, . cos ac". 2a: = 2a: cot ac”.
sin ac”
INTERCHANGE OF THE WARIABLES. 29
38. Interchange of the dependent and inde-
pendent variable. If in the theorem
du du dy
da, dy da;
we put 'll = 0,
du da (a + h) – a
then da, - da, * Lih-0 — — * l,
and we obtain the result
dy dº 1
da, dyT ‘’
dy I
OF da, -I: º
dy
39. The truth of this is also manifest geometrically,
dy da,
for 2 and ºf
da, dy are respectively the tangent and the co-
tangent of the angle J, which the tangent to the curve
y = f(a) makes with the w-axis.
40. This formula is very useful in the differentiation
of an inverse function.
Thus if we have y = f'(a),
w = f(y),
da, o,
and dy T f" (y),
a form which we are supposing to have been investigated.
Thus dy 1 1
dº TfG) ºf [f- (c)]
30 DIFFEREAWTIAJ, CA LOUI, U.S.
EXAMPLES.
Assuming as before for present purposes the following differen-
tial coefficients,
d a;8=33:2 d w | sin wa cos
— ſº - sº- –F– A/ Q) = —-– - Q = COS ſº
da: 2 da: 2 Wº’ da, 2
d I d -
— 6% = €” - log. a = - tan ac=secº a
da: y da, £e a ’ da, y
write down the differential coefficients of the following com-
binations:
1. e8°, e-", sinº a, Vsina, Wlog, w, Vtana, sin War.
2. esin *, (2 tan *, e”, ev*, C loge *.
3. log, sin w, log, tan w, log. V, log, wº.
4, sinlog, w, tan log, w, Wsinlog, w, V sin Wa, log, sin War.
5. logo V sin We, tanlog, sin e^*.
CHAPTER IV.
STANDARD FORMS.
41. IT is the object of the present Chapter to
investigate and tabulate the results of differentiating
the several standard forms referred to in Art. 21.
We shall always consider angles to be measured in
circular measure, and all logarithms to be Napierian,
unless the contrary is expressly stated.
It will be remembered that if u = b (a), then, by the
definition of a differential coefficient,
º - Lt.-0 q, (a. + º- q, (a) &
da,
42. Differential Coefficient of a ".
If w = p (a) = w”,
then q (a + h) = (a + h)",
du (a + h)” – a "
and da, T Lin–0. |-
(l + º) — I
= Lh-ow"—#—.
h
Now, since h is to be ultimately zero, we may consider
h
a. to be less than unity, and we can therefore apply the
32 DIFFERENTIAL CA LOU/, U.S.
e is hN*
Binomial Theorem to expand (l +.) , whatever be the
value of n ; hence
*–L, w" ( , h, , ), (n − 1) hº
in-ºn-0 || ". 2T ºf
u(n-1)(n-3) ** º
3 £3
h *
= Ltº-onw" | 1 + x (a convergent series)
{U
= }laſ”Tl.
43. It follows by Art. 35 that if u = [q, (a)]” then
*-nº ()
EXAMPLES.
Write down the differential coefficients of
1. a, wº, wº, a 19, w8, wº, wº, wº, Jaº.
} ––
w-Ha Woºd'
3. (aw-H b)”, aw”--b, (aw)”--b, a (a +b)", a' (a +b).
-- l
2. (w -- a)”, a"+ aft, w” + a”,
a;3 agº &# 3:5
4. I+4+;i-Fi-Fi-F#14 * * * * * *
5. a+b V. Vºivº, Watsº V; 3/a+a,
a Wºº Wa-Wa' Wa-Waſ' V a-w’ \/ a -w
aa”--ba;+ c
4-º-º-º-º-º-º-º-º-º-º-º-º-º-º-º-º-º- i |} ( )." Q e 1) || 0: q
6. #####, (4-0)" (, ; b), (x+d)^ſ(4-by.
44. Differential Coefficient of aº.
If w = p (a) = dº,
q, (a + h) = a”,
STANDARD FORMS. 33
du gºth — aft:
- and da, *Eº Lt.-0 — —
= a” log.a. [Art. 15.]
du
— o? — 23; — o?
COR. 1. If w = e”, d. 9 logee = e^.
CoR. 2. It follows by Art. 35 that if u = e^*), then
du /
da, = eb (3). ºp (a).
45. Differential coefficient of log, w.
If w = b (a) = logaa.,
ºb (a + h) = logo (a + h),
du loga (a + h) – logo a
and da, T Lin–0 h
l
- Ih-0; loga (l + º) te
Let # =z, so that if h = 0, 2 = Co.; therefore
}
... logae. [Art. 14.]
- * du 1 _1
COR. 1. If u = logea, da, Ta, logee = a
CoR. 2. And it follows as before that if
w = logº (a),
du d'(a)
then do * ºb (a) g
34 DIFFERENTIAL CA LOUI, U.S.
EXAMPLES.
Write down the differential coefficients of
g 62% + 63%
1. e”, e-8, e”, cosh w, Sinha', II. e-z'
2. log Wa, log (a +a), log (aw-b), log (aa”--ba;+c),
1 1 +a; l 1 +a;% l
Og T-a: } Og Taº? Ogic Cº.
3. 4 (e), @ (logº), [{(3)}, ſº (a+3)]", I(a+w)"].
4. e” log (w-Ha), afte", aft. e.", 2*, a" (degrees).
5. log (a + e”), e”--loga, e°/loga.
6. Cºlogº, log (aeº), loga".
46. Differential Coefficient of sin a.
If u = b (a) = sin w,
q (a + h) = sin (a + h),
du sin (a + h) – sin a
and da, £º Lt.-0 h
. h. ( º
2 sin – cos | a + .
= Lt 2 2
h = 0 h
sin
= Lt.-0 COS (a + º
l, F h 2
2
= cosa.. [Art. 11; I.]
47. Differential Coefficient of COS a.
If u = b (a) = cosa,
q (a + h) = cos (a + h),
dw cos (a + h) – cosa,
and do." Lt.-0- h
Q?
WTAM DARD FORMS.
3
k
. h
SII). - / -
= — Lt #in ſº tº
=-ºn-o-T- SII). (a 2
2
= — sin a.
COR. And as in previous cases the differential
coefficients of sin b (a) and cos (b (a) are respectively
* - cos b (a). (b' (a),
and – sin (b (a). (b (a).
EXAMPLES.
Write down the differential coefficients of
1. sin 2a, sin na', sin” ar, sin a ", sin War.
2. V sin Wa, log sin a', log sin Wa, esinº, evsinº,
3. sin” a cos” ar, sin” aſcos" ar, sin” (na”), eº sin bac.
4. sin a sin 2a, sin 3a, sin a . Sin 2aſsin 3a.
5. COs a cos 24: COS 3.3, cos” aa.. cost bar. cos" ca.
48. The remaining circular functions can be differ-
entiated from the definition in the same way. It is a
little quicker however to proceed thus after obtaining
the above results. º
e , , . Sin &
(i) If y = tan a cos a
dy #. (sina,). COs a — f (cos ar) sin a
da, T COS*a,
cos” a + sin” a
= − = secº w.
COS* (U
36 DIFFERENTIAL CALCULUS,
z e g = COS
(ii) If y =cot we'.
º - (— sin a j º i. Fº w (cost)__ COSec”a.
(iii) If y = sec a' = (cosa!)",
% = (–1) (cos º (cos a) = º: = Sec a tan ac,
(iv) If y = cosec a' = (sin a j-,
% = (–1) (sin w) * # (sin a y = — . = — COSec 3: COt, a.
(v) If y = vers a = 1 — cosa,
dy ...;
dic = S1 Il Q3.
(vi) If y = covers a = 1 — sin w,
*/ COS Ø.
da:
49. Differentiation of the inverse functions.
We may deduce the differential coefficients of all
the inverse functions directly from the definition as
shewn below.
For this method it seems useful to recur to the
notation of Art. 27 and to denote q, (a + h) by U.
50. Then if u = q. (a) = sin- a,
U = b (a + h) = sin- (a + h).
Hence a = sin u, and a + h = sin U :
therefore h = sin U – sin u,
du U — u U — w
and — = Lt = 0 —7– = Lt _2, --—FF---—-
da: h=0 T. *** sin Uſ – sin at
STANDARD FORMS.
3
7
U — u
2 *1
= Ltv-u == −
Sl]] -- 2 & CO 2
l I I
= — = --H = ~7+= }
cos w \ſi – sinºu MI - wº
and the remaining inverse functions may be differen-
tiated similarly.
51. But the method indicated in the preceding
chapter (Art. 40) for inverse functions simplifies and
shortens the work considerably.
Thus:–
(i) If w = sin- a,
we have w = sin u ;
da,
whence + = COS u :
dw
du 1 1. l 1
and therefore j = + = − = -7–– = +=-;
- da da cos u Ji – sinº, JIL aſ:
du
e T g
and since cos’ a = ; – sin-'a,
A-l
d cost" a. 1.
We have —H- = — — — .
da, V I — agº
(ii) If w = tian" as,
we have a = tan M ;
whence da, = secº w;
du
du 1 I 1
and therefore º-'gº-1 Etanºw TI-Hº
T
and since cot" a = | – tan" w,
2
38 D/FFERENTIAL OA LOUI, U.S.
d cot-1 a. I
we have -—H = -----, .
da, I + æ"
(iii) If w = secT'a,
we have = Sec u :
da:
whence + = Sec utan w;
du
* du cos” w I I
and therefore da, Tsinº, T TH - Jºi
Q, SII). QM, a: , /a:” – 1
*Mi-
&
º —] T
and since cosecT'a' = 2 T secT a .
d (cosecT'a, I
we have d (cosec wy F - -–7–– .
da, a Jay-T
(iv) If w = vers' ac,
we have a = vers w = 1 — cos w;
da, .
whence –F– = Sln M ;
du
du I l l
da, sin u VT-cos u J2a – wº
d covers "a, I
whence also -------- = - —7––.
da, 2a – acº
EXAMPLES.

Write down the differential coefficients of each of the follow-
ing expressions:
1. seca:4, sec−1 aſ”, tan wº, tan-1 aº, vers a;4, vers-1 a.”.
2. tan' e”, tane”, logtan ar, logtan T'a, log (tana)".
Q? ((? 1 — wº
3. Vers-1 : vers 1 (a + a tan-1 : cost! – .
4. Wooversa, tan” wº, (tan-1 wº)", a log tan-1 w.
5. tan & . sin T'a, secTitana, tan-' seca, e” sin-1 a.
STANDARD FORMS. 39
52. TABLE OF RESULTS TO BE COMMITTED To
MEMORY.
Q = 0;”. du = maſſ-l.
da,
du
'll, - d.º. * = a” log.a.
da, £e
dw
Q = €”. –F– = €”.
da,
logº. “-'love
Ql, F. '. ---— ` - g
§a dº - . 1982
lo dw 1
ºl, E (0. —H-- = — .
£e da, a
. Q!,
?!, F SII) (U. -- = COS 40.
da,
du e
Q!, E COS (U. –H– = —- SII). Q3.
da,
Q!, 2
w = tan a. –F– = SeC*@.
da,
du 2
Q = Cota. –7– = — COSéC* {\}.
da,
du sin a
Qſ, F SeC @, .7., T ... ... '
da, cos” a.
du COS ſº
Ql, E COSéC (0. —H-- = — — — .
da, Sin” as
w = sin-1 du I
F. (U. -- E —-
da, VI – wº
Q = COST! du I wº
-: 0). ---— sº – *-*
da, J 1 — aſ”
- —]. du I
Q = tian" a. —- = -— .
da; 1 + aſ
du I
Q = COt-1 a. — =
40 DIFFEREAWTIAL CALOUI, U.S.
w = SecT'a. du F. ––
da, a Jº — 1
'u = COSecT" as du = - I
da; Ø Vań – 1
w = VerST" a. du - I
da, V 2a – aº
Q = COverST" ºv. du - - ––
da, M2” – ſº
53. The Form w”, Logarithmic Differentiation.
In functions of the form u', where both u and v are
functions of w, it is generally advisable to take logarithms
before proceeding to differentiate.
Let y = u",
then logey = w logºu;
's-, 1 dy dw 1 du *
therefore g da, Tala, logow + * u dº? Arts. 31, 45,
or dy = ?!" (lo dw +" º
O - da, T 8° dº tº dº)
Three cases of this proposition present themselves.
I. If v be a constant and u a function of w, . = 0
and the above reduces to
dy , , du
U. M.”Tº -
da, da, ’
as might be expected from Art. 43.
STANDARD FORMS. 41
II. If u be a constant and v a function of a du 0
° da;
and the general form proved above reduces to
dy w” log, u do
da, T ge" dº
as might be expected from Art. 44.
III. If u and v be both functions of a, it appears
that the general formula
dy-, l dy , , du
da, QI oge” i. F Q)?!, da,
is the sum of the two special forms in I. and II., and
therefore we may, instead of taking logarithms in any
particular example, consider first u constant and then v
constant and add the results obtained on these supposi-
tions.
Ex. 1. Thus if y = (sina)*,
log y = a log sin a ;
1 dy.
therefore – º – log sin a +a; cota,
$/ da;
d * tº
and # = (sin a jº (log sin a + a cota).
Ex. 2. In cases such as y=a^+ (sin ac)*, we cannot take logarithms
directly. Let w =a^ and v=(sina)”.
dy du , du
Then da, * da, do *
|But log was a log v,
and log v = a log sin a ;
whence *-*{1+logº,
da:
div , . º
and i:-(sin a)*{log sin a + a cota:},
and ... # =a^{1+loga) + (sin a "log sin a + a cota).
The above compound process is called Logarithmic
differentiation and is useful whenever variables occur
42 DIFFERENTIAL CA LOUI, U.S.
as an index or when the expression to be differentiated
consists of a product of several involved factors.
EXAMPLES.
1. Differentiate wºn”, (sin-1 c)", a", 3:23,
2. Differentiate (sin wy” +(cosa)*, (tan wºº-Hw"*.
3. Differentiate tana x logº x e^xaºx Wº.
54. Transformations. Occasionally an Algebraic
or Trigonometrical transformation before beginning to
differentiate will much shorten the work.
(i) For instance, suppose

2a:
* –1 Tº
y=tan 1 – a 2
We observe that y = 2 tan- a ;
whence dy 2 o “
da; 1 + ac”
(ii) Suppose y=tan-" 1+a. &
1 — a.
Here g = tan- a -i-tan-1 1,
dy 1
and therefore da, Ti-Igº'
(iii) If $/ = tan
– 4 own –1 -- *T* – –- " – cos−1 ×2
we have y=tan II. T 4 ta Ira-T-3 cos”;
1 + 1 + æ2
. du — tº
~ ā-Jīā’
EXAMPLES,
Differentiate : --
— orº - fy º T-L 2 –
1. tan-1” *. 2. tan-14' 3%. 3 an-, Viº 1.
1 – 3a; g-H pay Q?
4. tan-1 == . 5, elogº. 6. Sec-1
WT-a” 1 – 23:2
STANDARD FORMS. 43
7. sec tan - 1 a 8, cos-it- 45-1 9, sin-1 (33 – 4a:3)
e M. £º º cos’ T-1. ... Slil º
Wa-w a –2\}
10. tan-1 I+a;3 11, cosºl (1–2a"). 12. log ſe (...) } e
55. Examples of Differentiation.
Ex. 1. Let y=AV2, where 2 is a known function of r.
Here 1/ =2},
dy – 12 - # – 1
and dz =#2 Tă = 2V2'
dy dy d2 *
whence da, - dź º da, 3. (Art. 35)
_ _1 d2
T 2 N/2 da
This form occurs so often that it will be found convenient to
commit it to memory.
Ex. 2. Let y = evedº.
d (evº) d (evº) d (Vootº) d (cota)
Here --- F T-- . —------—-- . ------- ––
da: d (Voot r} d (cota) da:
—ovº 2
- € * — . ( — COSéC* @).
2 Vcota. ( )
Ex. 3. Let y=(sina)"g" cot{e” (a+b+)}.
Taking logarithms
logy =log c. log sin a +log cot {e” (a + ba)}.
The differential coefficient of logy is 1 dy g
$/ da;
Again, log a . log sin a is a product, and when differentiated
becomes (Art. 31)
*lo sin a + lo 1
ØC g gº sin.cosº.
Also, log cot {e” (a + ba)} becomes when differentiated
I
cofº (a+bº)} : [– cosec” {e” (a+b+)}]. [e” (a + ba) + be”);
da &
#= (sin wy”. cot {e” (a + ba)} [. log sin a + cota: . log a
–2e” (a + b + ba) cosec 2 (º te
44 DIFFERENTIAL CA LOUI, U.S.
Ex. 4. Let y = x/a+-bºcosº (logº). Then
dy d Jay-Vºcosº (logº), d (a”- tº cosº (loga)} ... d (cos(log a)}
dº d'Ha*-ºcoş(logº), x -- {COS (loga)} × −i (loga.)
- d (loga.)
× - i.
=}{a”-b” cos” (log r)}~}×{–20°cos (loga)}×{-sin (loga)}x :
-- b” sin 2 (loga.)
22 May 53 cosº (loga)"
Ex. 5. Differentiate acº with regard to ac”.
Let ac2 = 2.
da:6
da: daº da da 5a."
Then di. Tº dº diz Tº dº 2,
da:
2- ; a;3.
56. Implicit relation of w and y. So far we
have been concerned with the case in which y is ex-
pressed eaſplicitly, i.e. directly in terms of w.
Cases however are of frequent occurrence in which
y is not expressed directly in terms of a, but its
functionality is implied by an algebraic relation con-
necting aſ and y.
In the case of such an implicit relation we proceed
as follows:–
Suppose for instance
w”-H ſ” = 3awy,
then 3a;”-- 3:y” % = 3a. (y + 4; % y
i.e. 3 (a!" – ay) + 3 (y” — aw) % = 0,
* * dº a’ – aſ
giving º - — yº – an e
PARTIAI, DIFFERENTIATION. 45
57. Partial Differentiation. It will be per-
ceived in the foregoing example that the expressions
3 (a!" – ay) and 3 (y” — aa) occurring are algebraically
the same as would be given by differentiating the ex-
pression a' + y”— 3aay first with regard to a, keep-
ing y a constant, and second with regard to y, keeping
a; a constant.
When such processes are applied to a function f(a), y)
of two or more variables the results are denoted by the
of of
symbols 㺠3, Thus in the above example
ºſ-aſ-a-
ºſ-soa-
and Øy T 3 (y” – aw).
This is termed partial differentiation, and the results
are called partial differential coefficients.
58. A general proposition. It appears that in
the preceding example
Qf , ºf dy_
i. it Oy i.-0.
ºf
g d/__0w
Ol da, - Of e
0y
This proposition is true for all implicit relations between
two variables, such as f(a), y) = 0.
Suppose the function capable of expansion by any
means in powers of a, and y, so that any general term
may be denoted by Aa:P/9.
Then f(w, y) = XAa:Py? = 0.
46 DIFFERENTIAL CALCULUS,
Then differentiating
S. (Aper- yº + Aa:Pg!/?-, %) = 0,
OI’ >Apaº-yº + (XA aerº-oº: = 0,
of , of dy
Ex. If f(x, y)=a^+aºy +y”=0,
we have |-ºrieſ,
of 4 2 .
j=º +3) 7
dy_ 52.44-4a:”y
- 㺠T a 44-399 e
EXAMPLES.
Find º in the following cases:
1. a "+y}=a^. 2. a "+y” – a ". 3. eV =&y.
4. log ay-a”--y”. 5. Acu. yº–1. 6. al-Fy”–1.
59. Euler’s Theorem.
If u = Aw"y” + Baºy” + ... = XAa"y?, say, where
a + 8 = a + 8' = ... = m,
to show that 2,04 + Ou F 77,74,
O at Way-º.
By partial differentiation we obtain
0\t
º-mºm. 10.- 8
Øa, T >Aaa"—'yń,
0w
** = S an 18-l
0y XA sº >
FULER'S THEOREM, 47
then Q} º: + y ; =>Aaa"y” + XAGaºyº
غ " " ºn
= XA (a + 8) wº
= n>Aa"y? = nu.
It is clear that this theorem can be extended to the
case of three or of any number of independent variables,
and that if, for example,
w = Aa"yºzy + Baº'yºzy -- ...,
where a + 8 + y = a + 8 + y' = ... = m,
tºp 0w 0w 0w
then will * 5.4 y # = nº
The functions thus described are called homogeneous
functions of the n” degree, and the above result is known
as Euler's Theorem on homogeneous functions.
EXAMPLES.
Verify Euler's theorem for the expressions:
.# # ??, %, I - * cº 3/
(***)(***) ºr, "sin;
EXAMPLES.
Find % in the following cases:
1. 1+a, 2. 3/= Ja + £. 3. 3/= Ja? + £4.
— ()” — a 2 2 — Ao 2
4. Aſ I — a 5 v=- & 6. wVºſa
1 + ºr Wi-Faº Waº—oº
T-z _1... a "+a;+1
9. Q =tan-1 (log w). 10. V-sin a ".
11. $/=sin (e”) loga. 12. Aſ =tan-1 (e”) log cota.
13. Aſ =log Cosha. 14, 3/= vers' log (cota).
48
DIFFERENTIAJ, CALCULUS,
15.
17.
19.
21.
23.
25.
27.
29.
31.
32.
33.
35.
39.
41.
43.
44.
46.
48.
=cottº (cosec ac). 16. V-sin" --— .
3/ ( ) 3/ WI-Eº
=tan-1 - *
$/ A/a.” – 1
smº -1 (* Han-1” \ . _4 cos’ &
3/=b tan (itan ..). 22, 4)
WTº
18, y=(sin-1 wym (cos−1(c)”.
. . 1 . . , a + b cosa,
* * — 1 -: -1 *-º-º-º-º-º-º-º-º-º-º-º-º-º-º-º-
3/=cos (asin !). - 24, 3/=sin b--a cosa!"
g–e" log(secº w8). 26. Aſ = e^* cos (btan-1 w).
3/=tan-1 (a^*. wº). 28, y=sec (log, Woº +3).
3/=tan-" a +} log #. 30, 3/=log(loga).
3/=log” (w), where log” means log log log...(repeated n times).
Wºła-VW-3 tan;
*Wºº log Wö-Fa– wºung
y=sin-1 (a Wi-º-Val VI-wº). 34, y=101".
gy – e’. 36. y=e”. 37. § – we’. 38. 3/=a".
3/=a^+ * s 40. y =(cota)* +(cosha)*.
gy=tan-1 (aº wº) M* 42. y =sin-1 (ethnº").
1+a;
- 7%, ... ???,
v-V(i+º £) (-in £).
y-tº-ſwºrº. 45 y-(º)"
l
3/=(cosa)*. 47. Ay=(cot-1 a.).
9.
3/= ( +.) +&lt;. 49. y =b tan-1 (; un- %).
Q? Cº., &
A.A.A./P.L.E.S. 49
50.
52.
54.
58.
60.
62.
63.
64.
65.
66.
67.
68.
69.
70.
71.
72.
tan y=e” sin w.
e!/ = (a + ba"); - ai
(baſſ);
a-gº-ºº. 55.
—- º'" 3/_
3/=a log a + ba
a"y"— (a: + g)”.
51. aw”--2/ay--by”= 1.
53. (cosa)" - (sin y)*.
w-y log wy. 56, 9/=w". 57. Aſ =w".
59. aw”--2hay--by”--2ga;+2fy-H c =0.
—1
61. y =e"Tºw log secº w8.
_1 a”-b”, ſp /a: , q ſºft . -
If !/=3 o:2+b^ {:}; -- #} 2 shew that when
2pſ, q+p
(a+b): in dy_ (a+b) ºf
,2} = (#) then will da, - (#) e
Differentiate logo w with regard to wº.
Differentiate (wº--aw-Ha”)"log cot; with regard to
tan-1 (a cos ba).
w g ; , , –1 e -
Differentiate wº"T** with regard to sin-1 a.
e e 1 + c2–1 .
Differentiate tanT1 VHº-1 with regard to tan-1 a.
Wi-Faż-i-W1–º
Differentiate
WI-Ha.2–WI–2
with regard to WI–a4,
e & I -
Differentiate sec-4 2.II with regard to WT – a 4.
Differentiate tan-1
* * l
with regard to sect' 22 – I'
* ... with regard to sin- 20,
1 — wº g º 1+2%
Differentiate wº log tan- a with regard to º
&:
Differentiate tan-1
to Go dº 2
prove a % = −4–.
da: 1 – 3) log &
E. D. C. 4
If y=a’’
§() DIFFERENTIAL CALCULUS,
_4 dy 1 zººs
73. *-ij. *ā-i, º, . J;
1 + to oo 1+i a
* 14...
l dy I
74. If y=a+; 11 1 prove.-: * I
* + 2 + to oo * + i + 1
• * * } & + ..
75. tº-vºn- w/in & + Vsinº Wetc. to co,
3/1=cos aſ 24/– 1).
76. If S,-the sum of a G. P. to n terms of which r is the
common ratio, prove that -
(r-1) *=0. – 1) S.–78, -1.
P l 6. /P l
*" as + ... + ...,
(C
.2 •3 r.
78. Given C=1+rcos 0+ º 26 º
.2 cº, 3 sin
and S = r sin 6+! º: 26 7 º 36
shew that cºs º-(c.48% cos 6 ;
28 & 20 ſo, was:
6.-S. =(0 +S*) sin 6.
79. If y=Sec 44, prove that
• dy 16t (1–t') Bººk ºf x_* * * * * ** Tº º’s
d; * (I-6AEIA)” where t=tan ar.
80. If y = e−" sec−1 (a V2) and 24+a;% = wº, find % in terms of
aſ and 2.
81. Prove that if a, be less than unity
– ºr “...+ ºr ad inf. = l
1 +a; ' 1 +a;% 1 +a;4 1 +a;8 ... ad int.--.
JAAMPLE'S. 5]
82. Prove that if a, be less than unity
1 — 23; 2a – 4a:" 40:8–83:7 1 + 2a:
d ... ad inf = +---.
IºHº" IºI. II.I.st 1+a;+a;”
83. Given Euler's Theorem that
(U. t &: s * si {: g
Ji, º, cos : cos * cos ... cos º-tºº 5
or) 2 22 23 2? (U'
1. a; 1 a; 1 & g I
prove ; tan 2 t; tan 22 + 23 tan 23 + ... ad inf. =z- Cot w,
-č l
and l secº + l Secº 3. +
2 & * --- 2 –4.
22 2 24 2 Secº, + ... ad inf = coSecº a
26 23 4;2"
84. Differentiate logarithmically the expressions for sin 6
and cos 6 in factors, and deduce the sums to infinity of the
following series
(a) I + I + → l + I +
62 -- T2 62 - 2%tr? 6% - 32T2 62 - 427-2 tº e º tº e
() tº twº-,+,+,+,+,+…
1?--w? ' 22--a;2 3?--a:3 42-1-a;2
I l |
(c) ºatºat ºat tº sº sº & E &
(*) Hits+ H2 + 1 +3+…
85. Sum to infinity the series
-----, +} +, +, −,
1+w 2 I-Lº 4 I Lºf 8 11.3%
86. If H, represent the sum of the homogeneous products
of n dimensions of a, ºſ, 2, prove
9//al,0A, 1,9//a_, ºr .
(a) J} ** 3/ by + 2 *...*=nſ, 5
OH, OH, , ØH.
() #4 ºr *=(a tº H. .
CHAPTER W.
SUCCESSIVE DIFFERENTIATION.
60. WHEN y is a given function of a, and % has
been found, we may proceed to differentiate a second
time obtaining f (#). This expression is called the
second differential coefficient of y with respect to a.
We may then differentiate again and obtain the third
differential coefficient and so on.
The expression d (#) is abbreviated into #).
p da, \da, dº) /
... dºy d %) ... .......... */.
* dº dº (... is Written 6:3 ° and so on.
Thus the several differential coefficients of y are
written
dº ’ da;2 da;3 ° da;" " ' ' '
They are often further abbreviated into
$/15 $/2, $/33. tº º $/m. tº gº tº
Ex. 1. Thus if y=a^, we have
gi-na",
!/2= n (n − 1) aſ”,
Aſs= n (n − 1)(n − 2) a "T",
and generally y, = n (n − 1)...(n - r + 1) aº’,
!/n-i-I = {/m--2 = ?/ ?--3 F ... = 0.
SUCCESSIVE DIFFERENTIATION. 53
Ex. 2. If g = tana,
y1=Sec”a = 1 +y”,
yg=2yy1=2 (y-º-y”),
ya-2 (1+3/*) yi = 2 (1 +4y”-- 3/*),
y1=2 (89 + 12y”) y1=8 (2) + 5 y”+3y”),
&c.
Fºx. 3. If y = (sin-la)”,
y1=2 (sin-la)|J1–a4,
... Squaring, (1 — wº) y1%=4y.
Hence differentiating, (1 – a 4) 2yilya–2aylº=4yi,
and dividing by 2/1, (1 — wº) /2 – wy1=2.
61. Standard results and processes.
The m” differential coefficient of some functions are
easy to find.
Ex. 1. If y = e^* we have yi = ae”, y, = a”e”,
* * * * * *
!,-a’e”,
CoR. i. If a = 1,
g = e”, y1= €”, y2 = €”, ... ... ſa-e”.
CoR. ii. y=a^2=e^*.*;
y1=(loga) e”,” =(log.a) a”;
ya-(log,a)” e”*=(log,a)*a*;
etc. = etc.
gy, = (logea)” e” log. a = (logo a)” a”.
Ex. 2. If y= loge (a + a);
(–1) (–2)
Wi-Fi: Us- T(z+a)” us--Tº: tº t w w tº tº
_(–1) (–2)(-3)...(– n+1)
71 - (a + a)”
(–1)”- (n-1)!
(a + a)”
(–1)” n!
l
CoR. If ſ/ ºw--a ; :/n 5 (via); H ſº
$/
54 DIFFERENTIAL CA LOULUS,
Ex. 3. If y = sin (aa 4-b);
1/1= a cos (a.a. -- b) = a sin (as tº +3) ;
£) ...s 2T
!/2=a^ Sin (**** #):
º 3
!/3 = a” sin (as tº -- #) ;
# tº $ tº ſº ſº g g g g g tº a tº 4 & 8 tº s e g º e g g tº a tº it tº e º ſº
yn = a” sin (as + b + ‘...) e
Similarly, if y=cos (aa 4-b),
!/n= a” cos (as + b +%) &
CoR. If a = 1 and b = 0;
º g 7?
then, when y = Slm ac, !/n = Słłl (, + *) ;
Tºtr
and, when ty = COS ar, y, = COS (** *) *
Ex. 4. If y=eº sin (ba;+ c) ;
gi-ae” sin (ba 4-c) + be" cos (ba: 4-c).
Let a = r cos p and b = r sin q},
so that *= a” + b% and tan *=}
and therefore y1=re” sin (ba; -- c + p).
Thus the operation of differentiating this expression is equivalent
to multiplying by r and adding q to the angle.
Thus yg=*e” sin (ba 4-c 4-24),
and generally !),= r^e” sin (ba, + c + mp).
Similarly, if y = e” cos (ba, +c),
ya-r"e” cos (ba, +c+m}).
These results are often wanted and the student should be able to
obtain them immediately.
Ex. 5. Find the mºh differential coefficient of sin” ac,
We have Q/ = sin’s- (3 sin a – sin 3a).
1. tº
Hence !/n=1 % S1}} (* +".) — 3” sin (3. + #): *
5
5
,8 UCCESSIVE DIFFERENTI.ſ TION.
Ex. 6. If y=sin” a cosła, find ya.
Here v=} sin”2a: cos •= (1 – cos4a:) cosa.
1 *
~ 16 (2 cosa – cos 3a – cos 5a),
1. 777ſ 777ſ \ . 717ſ
and n=15 } COS (***) – 3” cos (* + #) – 5” COS (ºr ++) º
EXAMPLES.
Find y, in the following cases:
l 1 * l ..?
1. ag-Eb 2. a — a 3. a ba, a + ba
aw-H b a;2 I •º-º-º-º-º-º-º-
5. C& + d' 6. a – a d 7. (c-Ea); * 8. Wº –H 0.
9. (n+a)-#. 10, log (alo-Fb)”.
11. 3/=sin & Sin 2a. 12, 3/=e” sin a sin 2a.
13. 3/=e” sin” w. 14, 3) =e” cos” bar.
15. Ayasin a sin 2a sin 3a. 16. y =e” sin” a cosła.
17. y =sin” a sin 2w. 18. y =e" sin” a sin 20.
62. Use of Partial Fractions.
Fractional expressions whose numerators and de-
nominators are both rational algebraic expressions
are differentiated n times by first putting them into
partial fractions.
a;2 a? 1.
Ex. 1. **C.T.W.INVEdjº (a-b)(a-c) a - a
b? 1. c2 1
(5-c)(5-a) g-5." Gra)(e–) -o
(see note on partial fractions Art. 66);
a? (–1)” n! b2 (–1)” n!
(a-b)(a-c) (ºr "Ú-c)(5ta) is tiyºri
c? (–1)” nº
(cº-a)(e–b) (c. cyrti'
-F
therefore ya–
56 DIFFERENTIAI, CA LOUI, U.S.
ac?
''' (r. 1)*(x+2).
To put this into Partial Fractions let a = 1 +2;
Ex. 2.
1 1 + 22 + 2*
then Jº -5L-
1 /1 52 4 22 © tº e
=#(; ; ; +; sº) by division
1 5 4 1
*āz; † 9: " , EIz
1. 5 4
73G-I); "W(ºil) " (I2).
(n+1)! (–1)”, 5n 1 (–1)” 4n. (-1)”
whence ** 3Giyº: tşūiyā tīā) i.
63. Application of Demoivre's Theorem.
When quadratic factors which are not resolvable
into real linear factors occur in the denominator, it is
often convenient to make use of Demoivre's Theorem
as in the following example.
1 1
Let ***Iaşi (Ed) (a – ta)
_ 1 1 1
T2a lar – a a Haſ
1. º, 1. 1.
* – f – | -
Then Jn-5. ( 1)" m HRH Girl e
Let a = r cos 0 and a = r sin 9,
whence r?= ac2+ a” and tan 0– e
H _(–1)” n! (cos 0–1 sin 6) "1 – (cos 0+ i sin 0)-º-l
€11C0. Jºi ( — t $ (c t Sill ) }
— 1)” m e
-'ºsino. 11).
- 7?,
=' * sin (n+1) 6 sin” 6,
(l
where 6 – tan-1 .
QC
NUCCESSIVE DIFFERENTIATION. 57
CoR. I. Similarly if y=
(~ +b)?--a”
(–1)” ml ..., -
th--is-sm (n+1) 6 sin” 9,
where 6–tan-i –“ –.
b -- as
&: (l,
COR. 2. If Q/ = tan-l a 3. th=ºlº 5
– 1)*-1 (m. – 1) . . cº
and 4; n = (-rº-h sin m6 sin” 6,
where tan 0–£-cot !/.
EXAMPLES.
Find the nº differential coefficients of y with respect to & in
the following cases:
I 1.
& + Cº. l
3 y=#logº. 4. W= ºria,
5. –––––. 6. v.--→.
& *~ (…Tº.) (3-5). & "T(º-Laz) (º-Etº)"
20: |
= t — I F. ---
7. /=tan I — w? " 8. Aſ a;2-Ha!-- I
ſº (??
9 /=a+...+1. * ***E*L*T*E.T.'
64. Leibnitz’s Theorem.
[Lemma. If a C, denote the number of combinations of m things
7 at a time then will
7?, C, + m Crºi Fº-l-1 Cr- •
Ol' pºrt |r-1 ºriº |r |n-r-1 |*, +, +
n+1
- ribºr Frt--1 Crºi ..]
58 DIFFERENTIAL CALGULUS,
Let y = ww, and let suffixes denote differentiations
with regard to w. Then -
3/1 - ºliº) –H 'll?)1,
y, = u,v -- 2wº, + wV, by differentiation.
Assume generally that
$/n = tºnſ) + nChun-ºh + nChun-ſº +...
+ nCrun—ºr + 77, Crºwn-r—"rºl +. & ..+ ?!?), e ‘º .(2).
Therefore differentiating
n0, nC,
7, S- 1
nCrº
+ tºm—rt/r4-1 { } + #3 & ..+ t!?),4-1
+n.0,
= unº-Fm+10:00 - nº.10 ºn–0, 4-mºGsun-gºs-H...
+ nr 0, fºun-rºl-H...+ war, by the Lemma;
therefore if the law (a) hold for n differentiations it
holds for m + 1.
But it was proved to hold for two differentiations,
and therefore it holds for three; therefore for four;
and so on; and therefore it is generally true, i.e.,
(w) = u,v -- Cºun—10, --,0sºn gº +...
+ nCºun-ru, +...+ wwn.
65. Applications.
Ex. 1. p = a "sin aa.
Here we take sin aa, as w and ar" as v.
Now v1-3a*, v2–3. 23, va–3. 2, and v, &c. are all Zero.
& 727ſ.
Also wn = a” sin (as + #) , etc.
2
Hence by Leibnitz's Theorem we have
& 777- * * – 1
!/m ==a:3a” sin (as + *) -- 713acła”-1 sin (as + Tº r)
SUCCESSIVE DIFFERENTIATION. 59
n (n − 1)
+-5
*-*-*. . 2.1a"-3 sin (as +*** ).
The student will note that if one of the factors be a power of a
it will be advisable to take that factor as v.
– 2
3. 2a:a??-? sin (as +*.* -)
Ex. 2. Let y = a 4. eº; find ya.
Here v=a:4, u-eº,
So that v1–4a:”, vo–12a:”, va–24a, v.4–24, and vº etc. all vanish.
Also w, = afte”, etc.
whence
q/g – afte”a:4+ 5a’e”. 4a:3+10. aše”. 12a2+ 10a264%. 24a: 4-5aea”. 24
=ae” {ała”--20aº-F120a2a2+240a2+120).
Ex. 3. Differentiate m times the equation
d”y dy
A-2 lº 2. <-ºr-ºw *
JC d;" *##y=0.
dº n (n − 1)
dºn (aºys)=&yn-la-Fºn. 22. Unli-F 2T 21/,
d?
dºn (ayi) = *}/w-H + 7ll/m,
dºy
dºn — !/m .
therefore by addition
a”y,42+ (2n+1) wyn H-H (n°4-1) yo–0,
* dº tºy div-Fl º/ dºy
Ol ac2 irº +(2n+1)*#+(*#1) i.-0.
Ex. 4. Even when the general value of y, cannot be obtained we
may sometimes find its value for a = 0 as follows.
Suppose y=ſlog (a + VI-Fº,
then gy, = 2 log (a + MT+3)/MIT tº * * * * * * * * * * * * * * * * * * (1),
and (1+a;”) y,”=4y,
whence differentiating and dividing by 2y,
(1 + æ") yº-i-tyi =2 ............ .............. (2).
60 DIFFERENTIAI, CA LOUI, U.S.
Differentiating m times by Leibnitz's Theorem
(1+aº) yakgº-2may, H+ n (n − 1) ya
+ æy, El-Hºmyn-0
Ol' (1+3*) y, Ł94 (2n+1) cy, H + m”y, =0.
Putting a = 0 we have
(!/wº)0= — m2 (Ja)0 is a tº e & • * * * * * * * * * * * * * * * g e º 'º e g (3),
indicating by suffix zero the value attained upon the vanishing of ar.
Now, when a =0 we have from the value of y and equations (1)
and (2)
(y)0=0, (/)0=0, (92)9–2.
Hence equation (3) gives
(/s)0=(VE)0=(!!?)0=...... = (yokº)0=0
and (yì)0= -2°. 2,
(ye)0– 4°. 2*. 2,
(ys)0= – 6°.4%.2%. 2,
etc.,
(yº)o- (-1)*-*2.2°. 4%. 6°......(2k – 2)”
= (–1)*-12%-1 (k-1)!}9.
EXAMPLES.
Apply Leibnitz's Theorem to find y, in the following cases:
1. 3/=ace”. 2. 3) =a^e”. 3. 3/=a” loga.
4. W =&ºsina. 5. 3/=e", sin bal. 6. /= it. *3
7. y = a tan-1 a. 8. 3) =wº tan-1 w;
66. NOTE ON PARTIAL FRACTIONS.
Since a number of examples on successive differentiation and on
integration depend on the ability of the student to put certain frac-
tional forms into partial fractions, we give the methods to be pursued
in a short note.
Let ; be the fraction which is to be resolved into its partial
fractions.
,SUCCESSIVE DIFFERENTIA TION. 61
1. If f(a) be not already of lower degree than the denominator,
we can divide out until the numerator of the remaining fraction is of
lower degree : e.g.
q2 3a – 2
G-jū-2) "Gijū-2).
Hence we shall consider only the case in which f(a) is of lower
degree than p (a).
2. . If p (a) contain a single factor (ac – a), not repeated, we pro-
ceed thus: Suppose
q, (a) = (a, - a) p (c),
and let _ſ(*) – -- 4 X (a)
- (cº-a) p(v)= x - a "W(º) '
A being independent of a.
f(a) . X (a)
= A + (a: – a º
**4+(*-*);
This is an identity and therefore true for all values of the variable
a: ; put a = a. Then, since iſ (a) does not vanish when a = a (for by
hypothesis iſ (a) does not contain a – a as a factor), we have
f(a)
A = “…}+.
!, (a)
Hence the rule to find A is, “Put a = a in every portion of the fraction
except in the factor a – a itself.”
Hence
Ex. 6) –tº–º-*. -- *-*.
e (a – a)(a – b) T a = b a - a b — a a ~ b
2 2
Ex. (ii) a’--pa;+ q _ dºpa-Ha 1.
(a – a) (a – b) (c-c) T (a – b) (a – c.) aſ a
b%+pb + q 1 cº-pc-q 1
(5-cyū-ſa) -i " (c-d)(e–) g-e
QC 1 2 : 3
*" (ii) (disjºia) = g(†) - giat 50:5
* ac2
Ex. (iv) (cº-a)(x-5) tº
Here the numerator not being of lower degree than the denominator,
we divide the numerator by the denominator. The result will then be
expressible in the form 1 + #, + #, where A and B are found as
2 2
before and are respectively #, and #, e
62 DIFFERENTIA/, CA. LCULUS.
3. Suppose the factor (a – a) in the denominator to be repeated
r times so that
p (a) = (a, - a)* 'p (a).
Put a – a = y.
Then f(a) f(a+1)
- p (a) y”y (a +y) '
or expanding each function by any means in ascending powers of y,
__40+4.1/+429*...
y” (Bo-H Bly -- Boy”--...)"
Divide out thus:–
Bo-Bay–F.40+4.1/+ ... Co-F Cly + Cay”+...,
etc.,
and let the division be continued until y” is a factor of the remainder.
Let the remainder be y”X (y).
º C C C C.- X (y)
Hence the fraction = 4 + ··· + *;--... ++} + +} +
ºf tyrºit ſº y "Wały)
Co C1 Ca
== (a – a)” -- (a – a)”-l + (a – dºt tº º tº
9r-1 x(c- a)
a – a ' | (a) '
Hence the partial fractions corresponding to the factor (a – a)” are
determined by a long division sum.
+
a;2
E X. Take g (p-1)*(x+1 ) *
Put a – 1 = y.
* 1.
Hence the fraction=;tº tº
713
2+y\142y+y" (#4 ºvtſy” – gº.
1 +%ly + y
#y + y”
#y + #y”
#y?
#!/*-F#y”
- #y”
Therefore the fraction
*.*.*.*.*. 1.
2y: " Ayā" sy" Sūry)
1. 3 l 1.
*
2(−1): " 4G-1)** 8G-I) - 8 (II):
FXAMPLES. 63
4. If a factor, such as a:*-i- aa. -- b, which is not resolvable into real
linear factors occur in the denominator, the form of the corresponding
Aac-i-B
rtial fraction is −– .
pa. 1O a"--aa. --b
For instance, if the expression be
1.
(a – a) (a – b)” (a.” + a”) (a.” + b”)”
the proper assumption for the form in partial fractions would be
A P C Da; + E Fa: + G : Ha; -- K
a — a a ~ b (c-b)* ' w” + a” a!” + b” (c2+b^)?’
where A, B, and C can be found according to the preceding methods,
and on reduction to a common denominator we can, by equating co-
efficients of like powers in the two numerators, find the remaining
letters D, E, F, G, H, K. Variations upon these methods will
suggest themselves to the student.
EXAMPLES.
Given y=sin a”, find yº, ys, WA.
Given y=a sina, find ºſs, ºys, /.
Given y=e” sin a, find /3, ...... 3/6.
Given y=a^e”, find y, and y.
If y=Ae”-- Be-", prove yº-m”y.
If y=A sin ma;+ B cos m.a., prove 3/2= — m/.
If y=a sin loga, prove aº/2+3/1+/=0.
(C
QC
-- Or -- - * ,3 9 = – ’) ' 2.
If y=log (i.º , prove wº/2=({ſ- ayi)
9. If $/= A (c-i-Wºº-I)” + B (a:- Wºº-i)",
prove (a”– 1).3/2+ ayi — mº/=0.
_(w- a) (a – b) ,
10. If 3/= (x-c)(x-d)? find yo.
11. If l
"Tºo-T);(ºr 2)” find ºn.
12. If 3/=a” logo, find Jº, V8, Va., W., 11.
13. If a = cosh (, log w) 5
prove (v*-1) /2+ ay! — m”y=0,
and (a" - 1)/, 2+(2n+1)ay, +1+(n°– m”)?),=0.
' ' ' ' + g_va (3–1) (I-71) w–1 ºvu =6) pub
‘’” — aſſo (g —w) (z–w) (I-w) u +z-vº (I – w) M – ºr=dſ
3.19UAA
‘. . .” *4-w Soo 8) + * +w UIS of = & ) wº
I + wº | .ſl/?!, $x \ y & .fi/?!, A ** a US w/9
18U1 eAO.I.I '83
‘quensUOO SI
#. g #. --- #. #. quuſ, eAOld
‘q=9 Soo 6–9 (Iſsa, put
‘p =9 ups ſº-H 9 Soo & JI "33,
ºpogº gº up-"ſº
‘(I-Fuj u + wouz-zººp) 200−/ JI “Tº
#(rug us (I–)+ I} ſu-"n
‘a’w Soo-Haru Uſs=77 JT '03,
ºp+ g?! = 4. 0–ºf pue
&
'o-"/ (go-Lºu) – it "ſºw (I-Fuz)—ºt "ſ' (za – I)
yeuſ, eAoûd 'rºus ,0=ſº JI '6I
'0–"ſ; (I +w) at +***ſ. I — a (I+u) z}+***# (ºr +I)
qeul axoid ‘r, una = ſº JI '8T
*/,
ºut-ºu-gº," "T
quul Aous oolio H
“ſ (ºu – gu)+++"ſw (1+uz)=***/ſ (ºr – I) pue
‘figu – Vſia –% (ºr – I)
‘(a r_uſs tu) UIS = ſº JI Tºul oAO.I.I ‘LI
* (I+2) t(I-9)—ſ. Jſ uſ pUII '91
gº
*- JI “ſº pull 'g'ſ
He- Jſ “ſº pIII “FT
'S'ſ) T/207 JTC) TFTW/AW.73/7/// (ſ #9
EXAMPLE'S. 65
24, Prove º
d” /COs & 70,7ſ º 77,7ſ. º
dº (*)- [P COS (ºr #)- o in (4 º)] | wººl,
where P and Q have the same values as in 23.
25. Prove that
Oft (e. SII). º =e” {Psin (bo-Frºb)+Q cos (ba-Hºnq})}/a^+1,
where * -
P=(ra)*—n (ra)*-* cos (p + n (n − 1)(ra)***cos 24 – ...,
Q= n (ra)* 'sin p-n (n-1) (ra)*-*sin 24 +...,
*=a^+b%, and tan q =b/a.
26. Prove that
#. (a "sina)=n (Psina;+ Q cosa),
2 4
where P=1 -"Cº- *C, º *
4:3 4.5
and - Q=” 137–"C's ăi-F"0, #1-....
27. Shew that
d" ſloga:
dºwn \ &m.
_ _ (-1)^n | T(m+n-1)! -?" (#}}]
T (m-1)! wºn tº * ! log & *, * ! (n-r) JJ
[I. C. S., 1892.]
CHAPTER WI.
EXPANSIONS.
67. THE student will have already met with several
expansions of given explicit functions in ascending in-
tegral powers of the independent variable; for example,
those tabulated on pages 10 and 11, which occur in
ordinary Algebra and Trigonometry.
The principal methods of development in common
use may be briefly classified as follows:
I. By purely Algebraical or Trigonometrical
processes.
II. By Taylor's or Maclaurin's Theorems.
III. By the use of a differential equation.
IV. By Differentiation of a known series, or a con-
verse process.
These methods we proceed to explain and exemplify.
68. METHOD I. Algebraic and Trigonometri-
cal Methods.
Ex. 1. Find the first three terms of the expansion of log seca:
in ascending powers of 2.
By Trigonometry
1 3:2 a.4 acó
COS (C: T; it 41 * GT-F.
Hence log Sec a' = -log cosa;= -log (1–2),
a;2 a.4 agó
where *=3 || - Tit GI - ... ;
EXPANSIONS. 67
and expanding log (1–2) by the logarithmic theorem we obtain
l 22 23
2 3:4 g6 1 Ta;2 a.4 -12
a;2 gº act,
* 3 T 51 t 720 T ".
34 0.6
+s. ~ 15 + …
3:2
hence log Sec w-g + 15 + 45 “
Ex. 2. Expand cos”a, in powers of ar.
Since 4 cosºa = cos 3a;+ 3 cosa;
7t,
= 1 – º – --
grºſſ-º,+(−1)" tº +
a-s mºº- - - – 1 \* —
© 1 a;2 a;4
we obtain cosa-i (1+3)–(3°48);14 (343) if-.
+(-1)*(**):...}.
1. a;3 a;5 a;7
e º - 3 = - 3 - - - *-*. -- - - -
Similarly sin *=1 |s 3); (35 3);I-(3. 3)# tº tº a
32n-1 – 3
- T \?? 7?–1 –
+(−1)"#Hijº º).
Ex. 3. Expand tana, in powers of a; as far as the term involving a ".
a;3 agº
* - 5 i + RT - ...
Since tan a' = g 5 #
1-#4 #:-
2 + 4 ‘’’
We may by actual division show that
tana: =a: a;3 2. 5
=a+5+15°4.
5–2
68 DIFFERENTIAJ, CA LOUI, U.S.
Ex. 4. Expand; log(1+z); in powers of .
Since (1+z)/=eylog(1+x),
we have, by expanding each side of this identity,
-- - – 2 - –9) al-
1+y+! (!, "1,2490-10-?),319 (9-90 °) (9-9). 1.
2 | 3 : , . . 4 |
2
=1+y log(1+x)+; (log(1+x)}*4.
Hence, equating coefficients of y”,
1 a 2 1 + 2 1. 2+2. 3+3. 1
- :\\2–1 — ºr orº 4 –
g(log(1+x)} =5i - -āj-º-F 4 | a;4 — etc.,
a series which may be written in the form
* - (1+!): 4 (1++++)* - (1++++++):
2 - \* + 2)-3 3 * 5) 4 (i+}++});4.
EXAMPLES.
1. Prove ***=1+2+! ****
e 3 * ' 120° ' ' ' '
774:3 4,4
2. Prove cosh” a = 1 + aſ +n. (3n-2) II ... .
sina, 4:2 art
3. Prove logº- T6 T T86 “ .
4. Prove logº*-, -i,....
5. Prove log & cot & = — * - ***
6. Prove log **. -** ; a:4– ###". -
7. Prove
42 20:3 4,4 & 5 a.0 ºf 4.8
- 2) — — - - - - - - - - - - - - - - - - - , , .
log (1–a7+a;%)= *+ 3 + -ā-H T- E - 5 7 + 5.
8. Expand log(1+a;"e") as far as the term containing a ".
EXPANSIONS. 69
9. Expand in powers of a,
–1 O-44, A so-1 *
(a) tan g-Epa, (c) sin I-Ea;2
3a – wº a; – 3:Tl
-1 — 1
(b) tan 1–34% (d) cos a;+a;T1 .
69. METHOD II. Taylor’s and Maclaurin’s
Theorems.
It has been discovered that the Binomial, Ex-
ponential, and other well-known expansions are all
particular cases of one general theorem, which has for
its object the eaſpansion of f(a + h) in ascending integral
positive powers of h, f(a) being a function of a of any
form whatever. It is found that such an expansion is
not always possible, but the student is referred to a
later chapter for a rigorous discussion of the limitations
of the Theorem.
70. Taylor's Theorem.
The theorem referred to is that under certain cir-
Cwm StarCéS
fº-foºlſ ()+...?"®-ºf"()+...
~ + º f" (a) +... to infinity,
an expansion of f(a + h) in powers of h.
This is known as Taylor's Theorem.
Assuming the possibility of expanding f(a + h) in
a convergent series of positive integral powers of h, let
2 3
ferº-4444444; +.......(1),
where A0, A1, A2, ... are functions of a, alone which are
to be determined.
7() D.I.FFERENTA.J. C.A.LOUI, U.S.
df (a + h) df (a + h) d(a + h) 2,
Novº-º; “º-f'(-r)
for w and h are independent quantities and therefore
a may be considered constant in differentiating with
regard to h, so that d º: h) 1.
Similarly
*ſº = f" (a + h); and so on.
Differentiating (1) then with regard to h, we have
/ df (a + h },2 }}
f (*)-º'-444/.44444 ...(2),
// df (a;+} h2 e
f (, ; )="ºtº- As + Ash +44...(3)
f*(*)-º- As + Aſh +...(4),
etc. = etc.
Putting h = 0, we have at once from (1), (2), etc.
A = f(a), A = f'(a), A, - f'(a), As = f'(a), etc.,
where f'(a), f" (a), f" (a)... are the several differential
coefficients of f(a) with respect to w. Substituting
these values in (1),
h” ..., hº
f(t+h)=f(x)+lf'(r)+; f"(t)+if"()+...
Ex. 1. Let f(a)=a.".
Then f'(a)=nº"-", f'(a)=n (n-1) wº, etc., and
f(t+h)=(a + h)”.
Thus Taylor's Theorem gives the Binomial expansion
(a + h)"-a" + mhaº-I-i- nº h2a:*-2+ ...
EXPANSIONS. 71
Ex. 2. Let f(z)=sin a.
Then f(x)=cosa, f'(a) = -sina, f'"(a)= — cosa, etc., and
f(a + h)=sin (a;+h).
Thus we obtain
2 3
sin (a + h)=sin *4 h cost-g inz- COS (C + ...
EXAMPLES.
Prove the following results:
I e”---herº •+”. €” +
& - 2 : 3 ! tº e º O & G. º
2. tanT' (a;+h)
h ach? 1 – 30:2 h9
=t — 1 a. -** = - - - - - - - - - - -
*T***IIº (II*) (II.5s 3
3. sin-1 (a;+h)
a hº , 14–23° hº
=sin T1 ac-i- -7---, + -ā ā-, + ... -- _ –
A/Taº (1–92); 2 (1–92); 3
4. Sec-1 (a + h)
=secT1 a + h__ 23%-1 hº + ... .
& Wºº-i 3.2 (9–1); 2
5. log sin (a + h)
=log sina;+h cot •-ºooºoºº- hº cosº.
2 3 sinº a:
71. Stirling’s or Maclaurin’s Theorem.
If in Taylor's expansion
fº-forºſ ()+...ſºrºr"()+...
We put 0 for a, and a for h, we arrive at the result
f(x)=f(0)+ºf'(0)+ºf"(0)+ºf"(0)+...
aft 70,
+ hiſ (0) +.. • 9.
72 JOIFFERENTIAL CALO/UI, U.S.
the meaning of f*(0) being that f(a) is to be differ-
entiated r times with respect to a, and then a is to be
put zero in the result,
This result is generally known as Maclaurin's
Theorem. Being a form of Taylor's Theorem it is
subject to similar limitations.
Ex. 1. Expand sin aſ in powers of a.
Here f(a) =sina, .." Hence f(0) =0,
f'(x) =cos w, . f'(0) = 1,
j" (a) = — sina, - f” (0) =0,
f". (a) = — GOS 3.3, f" (0) = — 1,
&c. &c.
f*(a)=sin (*#) º f*(0)=sin(;.
• ??,7ſ
Thus sin =*-*. agº ++,
Ul 111 (U = 4 à i + 51- “t * ! • * * *
Ex. 2. Expand log cosa; in powers of a.
Here f (a) =log cosa,
f'(a) = – tana:= — t, say,
f" (a) = -secº w = — (1+t”),
f"(
a)= -2 tan a secºa. = -2t (1+t”),
fº(a)= -2 (1+3t”) (1+t”)= -2 (1+4t2+3t"),
f(b)(a)= -2 (8t-i-12tº) (1+t”)= -2 (8t--20th-12th),
f(6)(a)= -2 (8+60t”4-60t") (1+t”)= -2 (8+68tº +120t" +60t"),
etc.
Whence
f(0)=log cos 0-log1=0,
and f' (0)=f(3)(0)= f(b)(0)=... =0,
also f"(0)= -1, f(9 (0)= -2, f(0) (0)=–16, etc.
Hence
2 4 6
* -2 Q} 16:-etc.
log cost--. IIT * Gl
EY PANSIONS. 73
EXAMPLES.
Apply Maclaurin's Theorem to prove
O 4 wn cost;
* @:
1. cost-1-H++,-,-,+–T++...
a;2 w8 a.4 w–1 *
2. log (1+z)=0 - 4 3 T ++...+(−1) ; +... .
4:3 4:5 32n-1
-1 or = 0: –– * --> — T Yn-1
3. tan T1 & = & 3 * 5 ... + (–1) 2. Tt
g T l
Sill 43 1. o.2 — ox! —
4, 6 =1+w-ga a *-... .
| l 24
20 \ – - 1. o.2 —
5. log(1+*)=log 2+3**** 192 "" " '
2 — hº 2– 2
6. ecosw-1+ax+** *******.
• sº
10.4% &” COS (a tan"l %) + . . . .
7, C!,
72. METHOD III. By the formation of a Differ-
ential Equation. First form a differential equation
as in Ex. 3, Art. 60, etc., and assume the Series
ao + ana, + (1,0} +...
for the expansion.
Substitute the series for y in the differential equa-
tion and equate coefficients of like powers of a, in the
resulting identity. We thus obtain sufficient equations
to find all the coefficients except one or two of the first
which may easily be obtained from the values of f(0),
f" (0), etc.
Ex. 1. To apply this method to the expansion of (1 + æ)”.
Let y = (1+a)”=ao-Fala + agº.”-- aga.” +...... ............ (1).
Then 91= n (1 + æ)” or (1 + æ) yi = my........................ (2).
But 91=a++2a2w-F 3a*-H...... .............................. (3).
74 DIFFERENTIAJ, OA LOUI, U.S.
Therefore substituting from (1) and (3) in the differential equation (2)
(1+z)(al-H2a3a; 4-3a*-i- ...) = n (ab-i- alº-Fasº.--...).
Hence, comparing coefficients
a1 =7tdo,
2a2+ al-na'i,
3a54-2a2=ma, etc.,
and by putting a;=0 in equation (1),
ao-1,
giving a1 = n,
m – 1 n (n − 1)
as--a-a--aſ-,
m – 2 n (n − 1) (n − 2
as--a- º, "ºº-º, etc.,
(l, _n - r +1 _n(n-1)...(n - r +1)
- r- r *—I - r! y
whence (1 + æ)”= 1 + ma;+ º a’--....
Ex. 2. Let y = f(a) = (sin-la:)”.
1.
= 2 Sin-la;. 3.
$/1 Vi — acº
(1 – a.”) yi”=4y.
Differentiating, and dividing by 2y, we have
(1 - wº) /2=wJi-H2................................. (1).
Now, let y=ao-i-ana -- aga.”--... + awa.”-- an Hºc” + and gº"+ ...,
therefore
y1=a+2age--...+na,” +(n+1) a, Ha*4-(n+2) alsº"+ ...,
and
y2=2a2+...+n(n-1)a,”+(n+1)na,11a*-*4-(n+2)(n+1)a, tºº"+ ....
Picking out the coefficient of a " in the equation (which may be done
without actual substitution) we have
(n+2) (n+1) an 12-m (n − 1) an=ma, ;
2
70, g
* g º º e º 'º e s tº s e º is a tº e s tº s e º a s is tº º (2).
therefore Cºn--2 = (n+1) (n+2) (ºn &
Jº YPAAWSIOWS. 75
Now, ao-f(0)=(sin-10)”,
and if we consider sin-1 a to be the smallest positive angle whose
Sune 1s ar,
sin-1 0 = 0.
Hence a0–0.
Again, a=f(0)=2in-o. ---0,
N/1 - 0
P/ 2
and ag=% f" (0) = } Hº-0 =1.
Hence, from equation (2), as, as, ay, ..., are each =0,
22 22 22
and *4 = 5.1. as–à i = i. 2,
4° __29.4° 22.4% 2
*S*5-6, 4=# TFTB =-BT 4.
etc. = etc.;
therefore
& , 242 22 2 . 42 2. 42. 62
(sin-1 *=. + 2. 2.44% ºf 2a:6-- *.*, *,2, sº s & © &
4 | 61 “ 8 |
A different method of proceeding is indicated in the
following example:–
Ex. 3. Let
g ſº a;2 a;3
y = sin (m sin-1 w)=do-Fahºrasji-Fasi: tº £ tº e º q & e < * tº s º º (1).
tº 7??,
Then gi-cos (m, sin" a.) —H. ,
VI-.”
whence (1 – wº) y,”= m” (1–y”).
Differentiating again, and dividing by 2/1, we have
(1-2%) ye-ay--m”y=0,.......................... (2).
Differentiating this m times by Leibnitz's Theorem
(1–3%) yaks – (2n+1) wyn H + (m”-n”) ya-0............ (3).
Now a0 = (y), 0– sin (m sin” 0)=0,
(assuming that sin-1 a, is the smallest positive angle whose sine is ar)
al= (y1)2–0=m,
a2=(J2)x-0=0,
etc.
an=(/a),—0.
76 DIFFERENTIAL CALCULUS,
Hence, putting a =0 in equation (3),
an 12- - (m”— n”) an.
Hence ag, ag, as , ..., each=0,
and ag= – (m”–1*) al- – m (m°–12),
ag= – (m?–3°) as – m (m2 – 1%) (m”–3°),
az= – (m”– 5%) ag- – m (m2–1°) (m°–3°) (m”–5°),
etc. º
Whence
2 – 12 2 – 12 2 – 22
sin (m sin-'a)=ma – nº a 3+” (m”- º (m?–3°) a;5
_m (m”–1*)(m”–3°)(m”–5°) a' + ....
71
The corresponding series for cos (m sin-1 a.) is
20.2 m2 (m2 – 92
cos (m sin-º)=1-ºr ***.
m” (m”– 2") (m.”— 4°) a;6-i-....
6 :
If we write z-sim 0 these series become
2 – 12
sin m6 = m sin 6 — nº Sin? 6
2 12 2 – 22
+” (m º (m?–32) sin” 0 – etc.,
_a mi"...s a , m*(m”–2°),
cos m0 = 1 – 3Tsin 6 + —II- Sinº. 9
2 ſm2 — 22) (m.2 — 42
m” (mº-2") (mº–4°) sinº. 6 + etc.
6 |
EXAMPLES.
1. Apply this method to find the known expansions of
a”, log(1+3), sina, tan T1 a.
2. If v=sin-ºw-a, -a, -oº-oº-H... 5
prove (1) (1–3%) 3/2=ayi,
(2) (n+1) (n+2)a, +2=n^a,
1 & 1. 3 a.5
(3) sin"' a' = a +;
ă ăt 3.4 . Hi-F....
EXPAAWSIONS. 77
3. If y=e^*=do--aw-Ha,”--agº--...,
prove (1) (1 -aº) y2=ayi-Faºy,
(2) (n + 1)(n+2) an 12=(n°4-a”) a, ,
ivo – a%2% a (a” + 1 a? (a^+2%
(3) gasm *–1+ax+ºf+ agº) *****).
?-- 1) (a 2-1-3?) .
+2 (a rºº + '*4. 3
(4) Deduce from (3) by expanding the left side by the
exponential theorem and equating coefficients of a, a”, a "... the
series for sin-1 a, (sin-1 w)”, (sin-1 a.)".
4. Prove that
(tan-1 wy?
~2.
a;2 I\ 0:4 I. I\ a,6 1 1 1 \ a,3
5. Prove that
1 ----------- a;2 2 a.4 2. 4 &6
- 2\l2=- – - - - -- - -
(a) [log(v4-VI+º]*=% – a . . . . . . . -...,
2
WI-Faż 3 . 5
73. METHOD IV. Differentiation or integra-
tion of a known series. The method of treatment
is best indicated by examples.
Ex. 1. If we differentiate the series
sin-º-º-º-º: acº 1. 3. 5 a.”
***, 3 + 3 + 3 + 3 + 67t.
we obtain the binomial expansion
1 1. 1. 3 1. 3. 5
——= 1 + · ac”-- - -a;4-- *|† : acº
JHa-'tarts:4°45.4.3°4.
and it is clear that we must be able by a reverse process (integration)
to infer the first series from the second.
The student unacquainted with integration may obtain the expan-
sion of sin-1 a from that of (1-wº)-à as follows:
78 DIFFERENTIAL CALGUI, U.S.
Let sin" was ad-i-aia, + aga.” + agº.”--...... !
then differentiating
1
H=a+2a,24-3a,”: 4a,”--.......
VI-.”
1 — 1.1.2, 4.3.3
H - 1.3
€1106 a1 =1, 2a2=0, 3a5-4, 4a3–0, 5a5–31, etc.
Also ao-sin-1 0=0
(if we take the smallest positive value of the inverse function).
Hence substituting the values of these coefficients
sin-º-º-º: 1. 3 a.”
=**3 g + 2.1 F +.......
Ex. 2. We have proved in Ex. 2 Art. 72 that
(sin-'a)*_aº 2*, 2*.4*.e., 22.49.6%
— — — ºr - -H -- *H* ––– 8
2 | gir IT +-gſ- "+-si-º-H
Hence differentiating we arrive at a new series
sin'" a tº as +*.*.*.*.*.*.*.*
Ji-i-"5 5 ! #1-44.
If we put ac-sin 6 we may write this as
20 22 . . 2 22.42 łv,4 22.42. 62
sing-1+5.1 sin 6 + 5 ! Sinº. 6+ 7
*...* 2.4 *4
Ol' =1+. SII). 0+; H SII). **ā-āī
sin" ()--...,
sinº. 6+....
EXAMPLES.
1. Obtain in this manner the expansion of
1 +3;
1 — a
log(1+w), tan'a, log
2. Prove
—s, 1 & 1.3 aft
2\ – - -
log (v4-W1+*)=& 3 3 + 3.4 5 - " '
EXAMPLE'S. 79
3. Expand
* 2 – A/1 2.2
º, an º-, ****
1+a; A/1-22 WI-Faż-- WT-aº
in powers of 3.
4. Prove
6 2
(i. ..)
22 22.42 o 22, 42.62
— 1 -i- - -- sin? —sinº —sinº
=1+aisin”0+5++-asin'6+ 3 +...+ssin"6+...
5. Prove that
easin-ºw
l *-ºs-º-º-º-º-º-º-º-
(1) VT-2
aa. , (a^+1})a? , a (a4+2%)a' , (a4+1%)(a4+3%) wº
=1+H + 2 : + 3 + 4 + ... ,
6)
€
(2) cos 6
= 1 sin 6 (1+12) sin” 6 (1 + 2*) sinº 6
--- 1: gi-F -s;
1 + 12 2\ cºrn 4
+ +1°) (1+3*) sin *....
4
EXAMPLIES.
1. Prove
1 2
log(1+tana)=w-5°43′4.
2. Prove
2 COS & *_ w” 110° wº
€ =1+w-H; 3 24 5 “ .
3. Prove
1 & 542 @3 251
- = -- ~ + → *
log {}; log (1+x) 24 8 + 2880 &’. . . .
4. Prove
a;2 24:3 aft aſ a 6 & 7
rºº 22 * = * *mºn * =s esses sºme = —- s—sº- &m-
log (1–a7+*) = — a + 2 + 3 * T ~ 5 3 7 +
80 DIFFERENTIAL CALCULUS,
w a;? II aft
5. Prove cosh (*cosa)=1 + º- 24 ' ' ' '
e 4:3 aſſº
sinh (woosa)=w-ºs - 5 ... .
2
6. Prove log *- #4. #~~
7. Prove
3 5
cos" (tanh log a = tr-2 {-, + º, - ~} e
8, Prove
Wi-Faż– l a w8 aſ aſ
tan-1 –––– = 3 — - + -— —-
8,Il & 3T G + IST I4
9. Prove log (33.4-4484-WI--9a2+24x4+163%)
I am? 1. 3 a.5 *
=3 {-} 5 t ...i. #-...}.
+ ... .
10. Prove that
3 7
1–wº sin-w-o-º- #: – † : * – ...
(a) (1–3%)? sin-1 w 3 3 5 3.5 7 y
sin
2
6
(b) 6 cot 6=1 —
7
@ 3-1-. ()–:, ; (;) -: , ()-
4 T * T 3 \2) 3 5 \2 5 - 7 (3) T“ .
11. Prove that
(w-E V1+º)"
2a-2 2 – 12 2 (on 2 – 92 2 — 12) (m2 – 22
+...
and deduce the expansions of 2
log (v--V1+3*), # {log (a +WI-Faº)}”, 3. {log (a + Wi-Faº)}.
12. If 3/=e” cosbal,
prove that 3/2–2a3/1+(a”--b%) y=0,
and hence that
2 — hº 2 — ºh?
68% COS w–1+a+*** **) a”--... .
EXAMPLES. 81
13. Prove
(a) sin (m tan-lay (1+3*)*
-º-º-º-º *** (*-1) (n-ºn-3 (m–4)
8.
a;6
- • * 3
(b) cos (m tan-1 w) (1+3*)?
* (m. – 1) o m (m. – 1 — 2 — 3
-1-" ºr 9.4 (m. * ) (m. '*- tº º 'º º
14 Deduce from 13(a)
tan- a log VI-Faż
I I\ a,3 I I I I\ a,6 | 1 1\ aſ
-(+)- (i+}+,+); +(41.4%) +- ... .
15. Prove . -
cosh 6 1* +1%, a (1%+1°) (1%+3°) ºf,
(a) COS 6 = 1 2 SII). 6-- –4– S}]] 6-H... y
sinh 6
COS 6
2 2 1 Q2 2 L O2\ / I 2 A 2
=} sin 6-- gº sing ****
(b)
sin”6+....
16, Prove as
tan-1 (a + h)=tan-4 a +(h sin 6) sin 6– * 6). sin 26
in A\3
+ * S
* 7, csirº AN4
in 36– * sin 46-i-etc.,
where a =cot 6.
17. Deduce from Ex. 16
2 3
(a) =0+cosésin 94. º 20+*. *in 20+...,
by putting h = — was — cot 6. -
Tr – 6 . . . 1 . I . 1 . .
(b) –3 – in 64; inser; in 204; in 104.
by putting h = – A/1-Haº.
(c) # = sin 6 , 1 sin 26 I sin 30, 1 sin 46
2 cos 6' 2 cosº. 6 3 cos; 9 4 cost 9 '''''
by putting h = – a – a “1.
E. D. C. 6
82 * DIFFERENTIAJ, CA LOUI, U.S.
18, Show that
I (sin-1a)?
(a) 2 Viº
4:2 I 1 \ wº I 1 1 \ a,6
= ~ + 12. 32 ſ :- -- ~ \ – 4-13 3% - 52 ſ --- it. ---it-
g-Hº. 3 (+ #); H . 3% - 5 (+ 33+ #) 61-F.-: ,
62
(b) sin 26
sin 6 1 1 \ sinº 6 I 1 1 \ sinö 6
- --> 2 I → -i- i.
+ ...
19, Prove
(tan-i º
T3 .
_l wº 1 I I\} agö (I. I. I
5, 3 T |}} (i+} + 1} 1+.)
I 1 TY) ºf
+}(44%)}; *
20, Prove
vers-1 & I , , i. 3 a.” 1. 3. 5 a.3
(*) = −1+3**śā āIt ####~
-1 on 2 1 4:3 1. 2 w8 1. 2. 3 4.4
* (vers’ º ºil ºf 1:2, ºr 1.2.3 °
(b) 2 ** 33 tº 3.5 - 3 + 3.5.7. It “
21. Prove that
h)+ f(a) — #2 8,
Mºtº/*-*-fºr; ſº f"().
22. Prove that
(a) f(ma.)
= f(x)+(m – 1) a f"(a)+(m – 1)* # "(w)+(m— iyºſ"()+ • * * 3
3:2
() f(#)
2 as
–ſº-H, fºrgº re-ai, #7"@4.
() fºr(O)4 af'()-ºf"ºf"º-ete
CHAPTER VII.
INFINITESIMALS.
74. Orders of Smallness.
If we conceive any magnitude A divided into
any large number of equal parts, say a billion (10”),
then each part is extremely small, and for all
I012
practical purposes negligible, in comparison with A.
If this part be again subdivided into a billion equal
parts, each = each of these last is extremely small
TO24?
in comparison with TOE’ and so on. We thus obtain
tº & A. A. A
a Series of magnitudes, 4, ion, 102, 1035” ‘’
of which is excessively small in comparison with the
one which precedes it, but very large compared with
the one which follows it. This furnishes us with what
we may designate a scale of Smallness.
., each
75. More generally, if we agree to consider any
given fraction f as being small in comparison with
unity, then fA will be small in comparison with A,
and we may term the expressions fa, f*A, f*A, ...,
Small quantities of the first, second, third, etc., orders;
and the numerical quantities f. f*, f*, ..., may be called
Small fractions of the first, second, third, etc., orders.
6–2
84 DIFFERENTIAL CALCULUS,
Thus, supposing A to be any given finite magni-
tude, any given fraction of A is at our choice to
designate a small quantity of the first order in com-
parison with A. When this is chosen, any quantity
which has to this small quantity of the first order a
ratio which is a small fraction of the first order, is itself
a small quantity of the second order. Similarly, any
quantity whose ratio to a small quantity of the second
order is a small fraction of the first order is a small
quantity of the third order, and so on. So that generally,
if a small quantity be such that its ratio to a small
quantity of the p” order be a small fraction of the q"
order, it is itself termed a small quantity of the (p+q)*
order.
76. Infinitesimals.
If these small quantities Af, Afº, Af", ..., be all
quantities whose limits are zero, then supposing f made
Smaller than any assignable quantity by sufficiently
increasing its denominator, these small quantities of the
first, second, third, etc., orders are termed infinitesimals
of the first, second, third, etc., orders.
From the nature of an infinitesimal it is clear that,
*f any equation contain finite quantities and infinitesimals,
the infinitesimals may be rejected.
77. PROP. In any equation between infinitesimals
of different orders, none but those of the lowest order need
be retained.
Suppose, for instance the equation to be
A, -- B, + C + D, + Ey + F + ... = 0...... (i),
each letter denoting an infinitesimal of the order in-
dicated by the suffix.
Then, dividing by A1,
B, , C, , D, , E, , F,
. . . . **2 L * 3 = (ii
At A + 4 + 4 + 4 + “ 0......(ii),
1 +
INFINITESIMALS. 85
the limiting ratios # and % are finite, while #. º 5
are infinitesimals of the first order, . is an infinitesimal
of the second order, and so on. Hence, by Art. 76,
equation (ii) may be replaced by
#4%-0.
and therefore equation (i) by
A, -ī- B -- C = 0,
which proves the statement.
1 +
78. PROP. In any equation connecting infinitesimals
we may substitute for any one of the quantities involved
any other which differs from it by a quantity of higher
order.
For if An -- B, + C, + D, + ... = 0
be the equation, and if A, = F +f,
f, denoting an infinitesimal of higher order than F, we
have F, -- B, -- C, +f, + D, -i-... = 0,
i.e. by the last proposition we may write
F, -ī- B, + C = 0,
which may therefore, if desirable, replace the equation
Al -i- B, + C = 0.
79. Illustrations.
(1) Since in e-º-º-º-º:
and coro-1-###-.
sin 6, 1 — cos 0, 6 — sin 6 are respectively of the first, second, and third
orders of small quantities, when 6 is of the first order; also, 1 may
be written instead of cos 0 if second order quantities are to be re-
jected, and 6 for sin 6 when cubes and higher powers are rejected.
86 DIFFERENTIAI, CALCULUS,
6
(2) Again, suppose AP the arc of a circle of centre O and radius
a. Suppose the angle AOP (=6) to be a small quantity of the first
order. Let PN be the perpendicular from P upon OA and AQ the
tangent at A, meeting OP produced in Q. Join P, A.
Then arc AP= aff and is of the first order,
NP:= a sin 6 do. do.,
A Q= a tan 0 do. do.,
chord AP=2a sim do. do.,
NA = a (1 – cos 6) and is of the second order.
So that OP-ON is a small quantity of the second order.

p_%
O N A
Again, arc AP– chord AIP = aff – 2d, sin
6 63
= a 6–2a (-sºrt.)
a63
*4.3 iT etc.,
and is of the third order.
PQ – NA = NA (sec 0–1)
. .. 6
2 sin.” 2
---
=(second order)(Second order)
=fourth order of Small quantities,
== NA .
and similarly for others.
80. The base angles of a triangle being given to be
small quantities of the first order, to find the order of the
difference between the base and the sum of the sides.
By what has gone before, (Art. 79 (2)), if APB be the
IWFIWITESIMALS. 87
triangle and PM the perpendicular on AB, AP – AM
P

–
A M B
and BP— BM are both small quantities of the second
order as compared with A.B.
Hence AP + PB— AB is of the second order com-
pared with A.B.
If AB itself be of the first order of small quantities,
then AP + PB— AB is of the third order.
81. Degree of approacimation in taking a small
chord for a small arc in any curve.
Let AB be an arc of a curve supposed continuous
between A and B, and so small as to be concave at each
P

_-s
A B
point throughout its length to the foot of the perpen-
dicular from that point upon the chord. Let AP, BP
be the tangents at A and B. Then, when A and B are
taken sufficiently near together, the chord AB and the
angles at A and B may each be considered small quan-
tities of at least the first order, and therefore, by what
has gone before, AP+ PB— AB will be at least of the
third order. Now we may take as an aariom that the
length of the arc AB is intermediate between the length
of the chord AB and the sum of the tangents AP, BP.
Hence the difference of the arc AB and the chord AB,
which is less than that between AP + PB and the chord
AB, must be at least of the third order.
88 JDIFFE/PEWTIAL CA LOUI, U.S.
EXAMPLES.
1. In the figure on page 86 suppose PM drawn at right
angles to AQ, and prove
(a) Segment cut off by AP is of the third order of small
quantities,
(b) Triangle PNA is of the third order,
(c) Triangle PQM is of the fifth order.
2. OAB is a triangle right-angled at A1 and of which the
angle at 0 is small and of the first order. A, B, is drawn per-
pendicular to OB, BA, to A, B, A,B, to OB, and so on.
Prove
(a) A, B, is a small quantity of the (2n − 1)" order,
(b) B.A., 11 is of the 2n” order,
(c) B,B is of the 2n” order,
(d) triangle BA, B, is of the (2m+2n − 1)" order.
3. A straight line of constant length slides between two
straight lines at right angles, viz. CAa, Cb13; AB, ab are two
positions of the line, and P their point of intersection. Show
that, in the limit, when the two positions coincide, we have
48 05 and £4_CAE
Bl, T CA "* PB = d.12.
4. From a point T in a radius of a circle, produced, a tangent
TP is drawn to the circle touching it in P. PW is drawn per-
pendicular to the radius OA. Show that, in the limit when P
moves up to A, AW
A = A.T.
5. Tangents are drawn to a circular arc at its middle point
and at its extremities; show that the area of the triangle formed
by the chord of the arc and the two tangents at the extremities
is ultimately four times that of the triangle formed by the three
tangents.
6. A regular polygon of n sides is inscribed in a circle. Show
that when n is very great the ratio of the difference of the
circumferences to the circumference of the circle is approximately
ºr?/6n2.
7. Show that the difference between the perimeters of the
earth and that of an inscribed regular polygon of ten thousand
sides is less than a yard (rad. of Earth=4000 miles).
EATAMPLE'S. 89
8. The sides of a triangle are 5 and 6 feet and the included
angle exceeds 60° by 10". Calculating the third side for an
angle of 60°, find the correction to be applied for the extra 10".
9. A person at a distance q from a tower of height p observes
that a flag-pole upon the top of it subtends an angle 6 at his eye.
Neglecting his height, show that if the observed angle be subject
to a small error a, the corresponding error in the length of the
pole has to the calculated length the ratio
qa coSec 6/(g cos 0–p sin 6).
10. If in the equation sin (o- 6)=sin a cos a, 6 be small,
show that its approximate value is
2 tan o sin': ( — tan” a sin” ..) e
[I. C. S.]
11. A small error w is made in measuring the side a of a
triangle, a small error $/ in measuring b, and a small error n”
in measuring C. Prove that the consequent errors in A and B
are each ºn", provided the relation
ba- an/ . &
:=} sin C-7, sin I"
be satisfied. II. C. S., 1892.]
2
CHAPTER VIII.
TANGENTS AND NORMALS.
82. Equation of TANGENT.
It was shown in Art. 18 that the equation of the
tangent at the point (a!, y) on the curve y = f(a) is
X and Y being the current co-ordinates of any point on
the tangent.
Suppose the equation of the curve to be given in the
form f(a), y) = 0.
It is shown in Art. 58 that
ôf
dy ºr
# = −7.
Öy
dy
Substituting this expression for ... in (1) we obtain
da,
TAMG ENT'S AND WORMALS. 91
A'ſ. (v_ººſ—
OI’ (X–); +() y); -o tº e º e º s º is e º º e (2)
for the equation of the tangent.
If the partial differential coefficients º 5 . , etc. be
denoted by fº, f, etc., equation (2) may then be written
(X – a f. -- (Y-y)f, a 0.
83. Simplification for Algebraic Curves.
If f(a), y) be an algebraic function of a and y of
degree n, suppose it made homogeneous in w, y, and 2 by
the introduction of a proper power of the linear unit 2
wherever necessary. Call the function thus altered
f(a), y, z). Then f(a), y, z) is a homogeneous algebraic
function of the n" degree; hence we have by Euler's
Theorem (Art. 59)
af. -- ºf +2.f. = nf(a), y, z) = 0,
by virtue of the equation to the curve.
Adding this to equation (2), the equation of the
tangent takes the form º
Yf. -- Yf, + 2f; = 0 tº e e º is e º e º 'º º e º is tº e º 'º (3),
where the 2 is to be put = 1 after the differentiations
have been performed.
We often for the sake of symmetry write Z instead
of 2 in this equation and write the tangent in the form
Xf, + Yf, + Zf. = 0.
Ex. f(x, y) = a ++ a”ay -- bºy-H cº-0.
The equation, when made homogeneous in ar, y, z by the introduction
of a proper power of z, is
f(x, y, z) = a ++ a”zyz*-i- bºyzº-i-cºzº—0,
and f. = 4a:*--a”yz”,
fy-a%22+b^23,
f, -2aºryz+3b*,224-46428.
92 DIFFERENTIAL CALOUI, U.S.
Substituting these in Equation 3, and putting Z = z = 1, we have for
the equation of the tangent to the curve at the point (a!, y)
X (43°4-a”y) + Y (a’a + b%)+2a2ay +35°y +4c4–0.
With very little practice the introduction of the z
can be performed mentally. It is generally more ad-
vantageous to use equation (3) than equation (2), be-
cause (3) gives the result in its simplest form, whereas
if (2) be used it is often necessary to reduce by substitu-
tions from the equation of the curve.
84. NORMAL.
DEF. The normal at any point of a curve is a straight
line through that point and perpendicular to the tangent
to the curve at that point. -
Let the axes be assumed rectangular. The equation
of the normal may then be at once written down. For
if the equation of the curve be
$) = f(a),
the tangent at (w, y) is
... — dy
and the normal is therefore
dy_
(X-w)+(Y-y).-0.
If the equation of the curve be given in the form
f(a), y) = 0,
the equation of the tangent is -
(X – a f. -- (Y – y) f, - 0,
and therefore that of the normal is
X— a Y-y
f. T f.
TANGENT'S A WD WORMALS. 93
Ex. 1. Consider the ellipse . + #= 1.
This requires 2° in the last term to make a homogeneous equation
in ac, y, and 2. We have then
a;2 g” 2
a;2 +. § -º = 0.
Hence the equation of the tangent is
2a: 2y
X. . 4-Y.; -2. 22=0,
where 2 is to be put = 1. Hence we get
: + #= 1 for the tangent,
and therefore *= = # $/ for the normal.
a2 W2
Ex. 2. Take the general equation of a conic
aw”--2ha y + by’--2ga;+2fy + c = 0.
When made homogeneous this becomes
aa”--2ha y + by” +2gaz-i-2fyz + cz”=0.
The equation of the tangent is therefore
X (aa. 4-hy +g) + Y (ha, + by +f) + ga' +fy + c = 0,
and that of the normal is
- X – aſ Y-y
aw--hy +g Tha + by +f'
Ex. 3. Consider the curve -log see: º
d
Then * =tant 3.
da: a.
and the equation of the tangent is
Y— y=tan (X-ax),
and of the normal
(Y-y) tan;4 (X-a)=0.
94 DIFFERENTIAL CALCULUS,
85. If f(a), y) = 0 and F'(a, y) = 0 be two curves
intersecting at the point w, y, their respective tangents
at that point are
Xf.4 Yſ, + Af. = 0,
and XF. -- YF, + ZF, - 0.
The angle at which these lines cut is
ſº-ſº
fºſ", + fiſh',
Hence if the curves touch
fº/F =f/F, ;
and if they cut orthogonally,
f. F., +f,F, = 0.
Ex. Find the angle of intersection of the curves
tanT1
a 3–3ay”- a,
32°y —yº–b.
Calling the left-hand members f and F respectively, we have
f.-3 (a.”— y”)=F,
f = – 6ay = – F.
Hence clearly J.F.--frºy–0,
and the curves cut orthogonally.
86. If the form of a curve be given by the equations
a = p (t), y = \le (t)
the tangent at the point determined by the third variable
t is by equation 1, Art. 82,
– ſo-Y90 –
or Xalº' (t) — Yp' (t) = q, (t) \,' (t) — ºr (t) p' (t).
Similarly by Art. 84 the corresponding normal is
Xºp'(t)+ Yº!'(t) = b(t) q'(t) + \l (t) lº'(t).
TANGENT'S AAWD NORMALS. 95
EXAMPLEs.
1. Find the equations of the tangents and normals at the
point (a, ºy) on each of the following curves:–
(1) w”--yº–c”. (5) a?y--ay”-aº.
(2) 3/*=4aw. (6) el/=sina.
(3) &y=k*. (7) w8–3avy--yº–0.
(4) 3/=c cosh ; # (8) (394-y”)}=a^(º-y”).
2. Write down the equations of the tangents and normals to
the curve y (a^+a”)=aa” at the points where y= º t
3. Prove that .4%=1 touches the curve y=be 8 at the
point where the curve crosses the axis of y.
4. Find where the tangent is parallel to the axis of a; and
where it is perpendicular to that axis for the following curves:–
(a) aa”--2hay--by”=1.
& – a 3
(3) y=
aw
(y) yº–wº (2a–w).
5. Find the tangent and normal at the point determined
by 6 on
(a) The ellipse a;= a cos 6 }
-*. 3/=b sin 6
(3) The cycloid a = a (6+sin }
3/=a. (1 – cos 6))
(y) The epicycloid we A cos 0– B cos # 6
3/=A sin 6— B sin #9
6. If p=3, cosa-i-y sin a touch the curve
aft", gy
i-F# =l,
777, º?, %
prove that pm-1=(a cosa)”-i-H(bsin a)7-1.
96 DIFFERENTIAJ, CALCULUS,
Hence write down the polar equation of the locus of the foot
of the perpendicular from the origin on the tangent to this
CUII*We.
Examine the cases of an ellipse and of a rectangular hyper-
bola.
7. Find the condition that the conics
aw” + by’=1, alwº-Fb'?=|
shall cut Orthogonally.
8. Prove that, if the axes be oblique and inclined at an
angle o, the equation of the normal to y=f(v) at (w, y) is
~ &\ , , v dy)
(Y-y) (coso.º.)+(x-º (l +cosº)=0
9. Show that the parabolas a "=ay and y”=2aw intersect
upon the Folium of Descartes w84-yº-3aay; and find the angles
between each pair at the points of intersection. - - -
87. Tangents at the Origin.
It will be shown in a subsequent article (124) that
in the case in which a curve, whose equation is given
in the rational algebraic form, passes through the origin,
the equation of the tangent or tangents at that point
can be at once written down by inspection; the rule
being to equate to 2ero the terms of lowest degree in the
equation of the curve.
Ex. 1. In the curve a”--y” + aa 4-by =0, aa. --by =0 is the equation
of the tangent at the origin; and in the curve (a^+y”)*= a” (a.”— y”),
a 2–y”=0 is the equation of a pair of tangents at the origin.
Ex. 2. Write down the equations of the tangents at the origin
in the following curves:—
(a) - (wº +y})*=aºtº – bºy”.
(3) aſ +y}=5aw”y?.
() (y-a). iſ "
b2,
TAMG ENT'S A WD WORMA IAS. 97
GEOMETRICAL RESULTS,
88. Cartesians. Intercepts.
e dy, v.
From the equation Y — y = da, (X – a j
it is clear that the intercepts which the tangent cuts off
from the axes of w and y are respectively
–4. _, dy
Q? # and v *d, *
da,
for these are respectively the values of X when Y = 0
and of Y when X = 0.
--~~
_^T O
Let PN, PT, PG be the ordinate, tangent, and
normal to the curve, and let PT make an angle J, with
the axis of a ; then tan le = %. Let the tangent cut
the axis of y in t, and let OY, OY, be perpendiculars
from 0, the origin, on the tangent and normal. Then
the above values of the intercepts are also obvious from
the figure.
E. D. C. 7

98 DIFF/ERENTIAL CA LOUI, U.S.
89. Subtangent, etc.
DEF. The line TN is called the subtangent and the
line NG is called the subnormal.
From the figure
Subtangent = TN = y cot lº =# º
da,
dy
Subnormal = NG = y tan ºr = y dº ’
Normal = PG = y sec * = y VI + tan” aſ

-vvi (;).
Tangent = TP = y cosec l = y wº
dy)*
V (%)
= y dy
da,
cos \le JI-F tanº, Vº %)
(;
•+y}
r -
OY = OG cos y = ON + NG da:

VI rº-VT,
These and other results may of course also be
obtained analytically from the equation of the tangent.
TANG ENT'S AAWD NORMALS. 99
Thus if the equation of the curve be given in the
form f(a), y) = 0,
the tangent Xf, + Yf, + Zf. = 0
makes intercepts – f. f. and – f.ſf, upon the co-ordinate
axes, and the perpendicular from the origin upon the
tangent is --
f//ſº +f*;
and indeed, any lengths or angles desired may be written
down by the ordinary methods and formulae of analytical
geometry.
Ex. 1. For the “chainette’”
6 : * –%
y=;(e're ‘)
1, 2 –3.
we have th=3 (e.” – e “).
- 2 - 2
C C
Hence Subtangent=# =cºtt.
1/1 3. 2.
e 9 – € C
23: 2a:
Subnormal-y- (e” – e º ).
----- 2
Normal-y Ji-Hyº-º, etc.
Ex. 2. Find that curve of the class ! = ani whose subnormal is
constant.
a;n-l
Here $/l = n gº-l y
a;2n-1
and subnormal = y/i = n ºn-3
Thus if 2n = 1 the ac disappears and leaves
0.
subnormal– 2 ”
and the curve is the ordinary parabola
g°=aac.
100 DIFFERENTIAL CA LOULUS,
ds da;
dº ’ de ’ “
Let P, Q be contiguous points on a curve. Let
the co-ordinates of P be (a, y) and of Q (a + 8a, y + 8/).
90. Values of to.

Y
Q
_*-*
A
O M N X
Then the perpendicular PR = 8a, and RQ= Sy. Let
the arc AP measured from some fixed point A on the
curve be called s and the arc AQ =s+ 8s. Then arc
PQ = 8s. When Q travels along the curve so as to
come indefinitely near to P, the arc PQ and the chord
PQ differ ultimately by a quantity of higher order of
smallness than the arc PQ itself. (Art. 81.)
Hence, rejecting infinitesimals of order higher than
the second, we have
8s” = (chord PQ) = (8a; +8/),
* _r, (8w”, 8y”) (da)* , /dy)"
OY 1 = Lt (...+ ...) *- (#) + (%) ſº
e - 8s” 8y?
Similarly Lt Šaj. T Lt (l + ...) 3.
* ds \*_ dy)* .
Ol (#) = 1 + (%) 2
and in the same manner
. . . ds \*_ da:\*
::::... (...) = 1 + (...) o
TAAG ENT'S AND MORMALS. 101
If ſº be the angle which the tangent makes with
the axis of a we have as in Art. 18,
anº-Lº. -Lº-º:
and also
PR PR 8a; da;
** = Pºiro–Pºo-º-;
and
& *º- RQ BQ r, 8./ dy
sin W = Pºiro–Lºſ-Tº-ji.
IEXAMPLES.
1. Find the length of the perpendicular from the origin
on the tangent at the point a, y of the curve
a:4+y}=64.
2. Show that in the curve y=be" the subtangent is of con-
stant length.
3. Show that in the curve by?=(a + a)3 the square of the
subtangent varies as the subnormal.
4. For the parabola y”=4aw, prove
ds /a+º
da, T A/ a
2 a.2
5. Prove that for the ellipse . + # = 1, if w= a sin q,
ds ——-
—— = – 22 2
dq, a VI-69 sin q).
6. For the cycloid a = a vers 6
y=a. (6+sin 6) J
}I’OVG ds 2a. {e
I da, T &
7. In the curve 3/=a log sec . 3.
ds ..??
ds J}
prove ... =See a dyT coseca, and w=aV.
102 DIFFERENTIAL CALCUI, U.S.
8. Show that the portion of the tangent to the curve
*4. yº– a",
which is intercepted between the axes, is of constant length.
Find the area of the portion included between the axes and
the tangent.
9. Find for what value of n, the length of the subnormal
of the curve a y”=a^*1 is constant. Also for what value of n
the area of the triangle included between the axes and any tan-
gent is constant.
10. Prove that for the catenary y=c cosh ; the length of
the perpendicular from the foot of the ordinate on the tangent
is of constant length. -
11. In the tractory
c– Woº-yº
prove that the portion of the tangent intercepted between the
point of contact and the axis of a, is of constant length.
w = Woº- 74. log
91. Polar Co-ordinates.
If the equation of the curve be referred to polar co-
ordinates, suppose 0 to be the pole and P, Q two con-
tiguous points on the curve. Let the co-ordinates of P
and Q be (r, 6) and (r-i- 8r, 6 + 86) respectively. Let
PN be the perpendicular on OQ, then NQ differs from

TANGENT'S AND NORMALS. I ()3
8r and NP from rô6 by a quantity of higher order of
smallness than 80. (Art. 79.)
Let the arc measured from some fixed point A to P
be called s and from A to Q, S + 8s. Then arc PQ = 8s.
Hence, rejecting infinitesimals of order higher than the
Second, we have
8s? – (chord PQ) = (NQ? --PN*) = (3rº +7°86°),
and therefore
(#) ºr(ſ)=1, or (*)-1+...(?)
*Y-re. %)
OI’ (...) = r (i. 5
according as we divide by 8s", 8r", or 80° before proceeding
to the limit.
92. Inclination of the Radius Vector to the
Tangent.
Next, let q be the angle which the tangent at any
point P makes with the radius vector, then
d6 dr . roló
tan b-ri, cost = i, , sin b-;.
For, with the figure of the preceding article, since, when
Q has moved along the curve so near to P that Q and
P may be considered as ultimately coincident, QP be-
comes the tangent at P and the angles OQT and OPT
are each of them ultimately equal to b, and
NP_r, rö6 ...d6,
tan – Litan NQP=Lºw-It-r;
NQ
cos ?– Lºcos NQP-Italiºn
NQ r, 8r dr.
= Pºp-Tº: -;
104 JDIFFERENTIAL CALOUI, U.S.
in º-Lusi Nop-L ºn
sin (p = Lt sin T“chord QP
NP r80 rdó
= Pºp-Tº: -;.
Ex. Find the angle p in the case of the curve
r”= a” sec (n0+a),
and prove that this curve is intersected by the curve
r”-b" sec (n0+ 8)
at an angle which is independent of a and b. [I. C. S., 1886.]
Taking the logarithmic differential,
*=tan (n0+ a),
Tr
whence 2 T q = m3+ a.
In a similar manner for the second curve
Tr ’ —
2 p' =n0+ 8,
q,' being the angle which the radius vector makes with the tangent to
the second curve. Hence the angle between the tangents at the point
of intersection is a ~ 6.
93. Polar Subtangent, Subnormal.
Let OY be the perpendicular from the origin on the
tangent at P.
Let TOt be drawn through 0 perpendicular to OP
t
and cutting the tangent in T and the normal in t Then

TANGENT'S A VD VORMALS. 105
OT is called the “Polar Subtangent” and Ot is called the
“Polar Submormal.”
It is clear that
/ , d6
OT -: OP tan q, - 7.2 dr te e º e º e º & 6 e º e º 'º a (1),
and that ot-opolº....…(?)
94. It is often found convenient when using polar
g ... 1 1 du
co-ordinates to write for r, and therefore — vºd for
Q0
d6
dr * e &
ja. With this notation,
d6
* _, d6 d6
Polar Subtangent = , dr du
Ex. In the conic lu = 1 + c coS 6
we have l=-esino".
dw
Thus the length of the polar subtangent is l|e sin 0.
Also, from the figure, the angular co-ordinate of its extremity is
0–.
2
Hence the co-ordinates of T (r1, 61) satisfy the equation
r1=lle sin (, -- º) t
The locus of the extremity is therefore
lu = c coS 6 ;
that is, the directrix corresponding to that focus which is taken as
Origin.
95. Perpendicular from Pole on Tangent.
Let OY = p.
Then .* p = r sin (b,
I06 DIFFERENTIAL CALCULUS,
and therefore
1 1 a 4 – • A-" *(i)}
;==º * ==(1+cot 9-#} + (i. }
7.2
1 I 1 /dr Y2
therefore pº == + r; (i) C & e º e º 'º e º e º e º 'º e º 'º - (1)
du\?
= 'wº + (...) tº is tº e e º 'º - e. e. e. e. e. e. e. e. e. e. e. e. (2).
tº 62
Ex. In the spiral "="; Li
we have aw- 1 – 6-2,
dw
lºº. – 9/9-3 .
whence (l, i;=20 y
and therefore, Squaring and adding,
a?
pº 1 – 29–24-6-44-40–6.
Thus, corresponding to 6 = + 1, we have
a? Oſ,
==4 and p =+.
96. The Pedal Equation.
The relation between p and r often forms a very
convenient equation to the curve. It is called the
Pedal equation. -
(1) If the curve be given in Cartesians,
say F (a, y)=0 ..................... (1),
the tangent is
XF, + YF, + ZF, =0
2 — F.”
and p =riº, “….(?)
If a, y be eliminated between equations (1), (2) and
a” + 3/* = * ........................ (3),
the required equation will result.
TANGENTS AND WORMALS 107
Ex. If a "+y}=2aa,
X (a, - a) + Yy=aa,
is the equation of the tangent, and
2 a2a:3 1 7'4
p = 7–Vä––3 = 7 —; ,
(a – a)*-ī-y” T 4 a.”
Or r°–2ap.
This result will also be evident geometrically.
(2) If the curve be given in Polars we may first
obtain p in terms of r and 6 by Art. 95, and then
eliminate 6 between this result and the equation to the
CUll"Vë.
Ex. Required the pedal equation of r"= a” sin m6.
By logarithmic differentiation,
m dr
7 déT m cot muff,
... cot p = cot mb or p = mt),
Q '???,
whence p = r sin ºp-r sin m{} = r dºm'
Or pa”—rm-H.
EXAMPLES.
1. In the equiangular spiral r-ae”, prove
dr º
+ =cosa and p =r sin a.
ds
2. For the involute of a circle, viz.,
2.T.,2
W. — C, O,
6 = } — cos−1 : ,
Ot, q'
Cº,
prove cos b-; e
3. In the parabola *-i-cos 6, prove the following re-
sults:—
() —-3.
108 I)IFFERENTIAL CA LOUI, U.S.
(l,
(8) p = T. º
* 3
(y) p” = ar.
(6) Polar subtangent=2a coSec 6.
4. For the cardioide r- a (1 — cos 6), prove
@ 4.
(8) p = 2a sin” *
7.3
*= -
(y) 20, e
sinº:
(6) Polar subtangent=2a.
COS 2
97. Maximum number of tangents from a
point to a curve of the nº degree.
Let the equation of the curve be f(a), y) = 0. The
equation of the tangent at the point (w, y) is
Xf, + Yf, + Zf. = 0,
where 2 is to be put equal to unity after the differen-
tiation is performed. If this pass through the point
h, k we have
hf + kf, +f = 0.
This is an equation of the (n − 1)" degree in a and
y and represents a curve of the (n − 1)" degree passing
through the points of contact of the tangents drawn from
the point (h, k) to the curve f(a), y) = 0. These two
curves have n (n − 1) points of intersection, and there-
fore there are n (n − 1) points of contact corresponding
to n (n − 1) tangents, real or imaginary, which can be
drawn from a given point to a curve of the m” degree.
TAAWG ENT'S AAWD NORMALS. 109
Thus for a comic, a cubic, a quartic, the maximum
number of tangents which can be drawn from a given
point is 2, 6, 12 respectively.
98. Number of Normals which can be drawn
to a Curve to pass through a given point.
Let h, k be the point through which the normals are
to pass.
The equation of the normal to the curve f(a), y) = 0
at the point (w, y) is
X – a Y-y
f f,
If this pass through h, k,
(h – a f, - (k - y) f.
th
This equation is of the m" degree in a and y and
represents a curve which goes through the feet of all
normals which can be drawn from the point h, k to the
curve. Combining this with f(a), y) = 0, which is also
of the n" degree, it appears that there are nº points of
intersection, and that therefore there can be m” normals,
Teal or imaginary, drawn to a given curve to pass through
a given point.
For example, if the curve be an ellipse, n=2, and the number of
!/
l)2 = 1 be the equation of the curve, then
a;2
normals is 4. Let aſ +
7
(h – wº #=(k-1)
is the curve which, with the ellipse, determines the feet of the normals
drawn from the point (h, k). This is a rectangular hyperbola which
passes through the origin and through the point (h, k).
The student should consider how it is that an infinite number of
normals can be drawn from the centre of a circle to the circumference.
99. The curves
and (h – a f, - (k - y) f = 0............ (2),
110 DIFFERENTIAL CALCULUS,
on which lie the points of contact of tangents and the
feet of the normals respectively, which can be drawn to
the curve f(a), y) = 0 So as to pass through the point
(h, k), are the same for the curve f(a), y) = a, And, as
equations (1) and (2) do not depend on a, they represent
the loci of the points of contact and of the feet of the
normals respectively for all values of a, that is, for all
members of the family of curves obtained by varying a
in f(a), y) = a in any manner.
EXAMPLES.
1. Through the point h, k tangents are drawn to the curve
Aa3+ By}=1;
show that the points of contact lie on a conic.
2. If from any point Pmormals be drawn to the curve whose
equation is y”=maa”, show that the feet of the normals lie on a
conic of which the straight line joining P to the origin is a dia-
meter. Find the position of the axes of this conic.
3. The points of contact of tangents from the point h, k to
the curve w8+y}=3ary lie on a conic which passes through th
Origin. -
4. Through a given point h, k tangents are drawn to curves
where the ordinate varies as the cube of the abscissa. Show that
the locus of the points of contact is the rectangular hyperbola
2ay+ka:-3ky–0,
and the locus of the remaining point in which each tangent cuts
the curve is the rectangular hyperbola
ay–4ka, +3hy=0.
EXAMPLES.
1. Find the points on the curve
gy–(a – 1) (a, -2) (a, -3)
at which the tangent is parallel to the axis of w.
Show also that the tangents at the first and third inter-
sections with the a-axis are parallel, and at the middle inter-
section the tangent makes an angle 135° with that axis.
EXAMPLE'S. - 111
2. In any Cartesian curve the rectangle contained by the
subtangent and the subnormal is equal to the Square on the
corresponding ordinate.
3. Show that the only Cartesian locus in which the ratio
of the subtangent to the subnormal is constant is a straight
line.
4. If the ratio of the subnormal to the subtangent vary as
the square of the abscissa the curve is a parabola.
5. Show that in any curve
Subnormal (i. ormal y
Subtangent T \Tangent
6. Find that normal to
W ay = @ + æ,
which makes equal intercepts upon the co-ordinate axes.
7. Prove that the sum of the intercepts of the tangent to
Wa;+Wy=Wa
upon the co-ordinate axes is constant.
8. Show that in the curve
3/= a log (a"—a”),
the sum of the tangent and the subtangent varies as the product
of the co-ordinates of the point.
9. Show that in the curve
ſm? -- ?? — Av??? - ??a,2m.
w”-a”- “y”,
the m” power of the subtangent varies as the m” power of the
subnormal. .
wº
10. In the curve y”=a^-1a, the subnormal oc º and the sub-
tangent oc a.
11. Show that in the curve y=be & the subtangent varies as
the Square of the abscissa. -
12. If in a curve the normal varies as the cube of the ordinate,
find the subtangent and the subnormal.
13. Show that in the curve for which
S=clog g
3/
the tangent is of constant length.
II2 DIFFERENTIAL CALCULUS,
14. Show that in the curve for which
3/*=c”--s”, (The Catenary)
the perpendicular from the foot of the ordinate upon the tangent
is of constant length.
15. Show that the polar subtangent in the curve r=a& (The
Spiral of Archimedes) varies as the square of the radius vector,
and the polar subnormal is constant.
16. Show that the polar subtangent is constant in the curve
º *6 =a. (The Reciprocal Spiral.)
17. Show that in the curve
r=ae”, (The Equiangular Spiral.)
(1) the tangent makes a constant angle with the radius
vector;
(2) the Polar Subtangent=r tana;
the Polar Subnormal = r cot a ;
(3) the loci of the extremities of the polar subtangent,
the polar subnormal, the perpendicular upon the tangent from
the pole are curves of the same species as the original.
18. Show that each of the several classes of curves (Cotes's
‘Spirals)
r=ae”, rô= a, r sin nô– a, r sinh mó–a,
7' cosh mó = a,
have pedal equations of the form
1 A
==#| ||
where A and B are certain constants.
19. Find the angle of intersection of the Cardioides
r=a. (1+cos 6),
r= b (1 — cos 6).
20. Find the angle of intersection of
wº–y?=a”
EXAMPLES. 113
21. Show that the condition of tangency of
- a cosa-i-ysin a = p,
- & - +
with any” = @” º,
is p” + 7t, & 770% & 70% - (m. + n)” + *Oſm -F 77, COS” (i. sin” Ol.
Hence write down the equation of the locus of the foot of the
perpendicular from the origin upon a tangent.
22. Show that in the curve (the cycloid)
a = a (6+sin 6),
$/= a (1 – cos 6),
ds
6 ds ---------- -
d6 =2a cos; and º-Way.
23. Show that in the curve (an epicycloid)
a = (a+b) cos 6– b cos º 6,
g . a+b
4/= (a+b) sin 6–b sin +0.
we have
--- i., " a . a +20 a. --- 3, alº .
p=(a+2') sing,0; Wr = 2b 6; p=(a+2)sin is
and that the pedal equation is
24. Show that the normal to y”= 4aa, touches the curve
27ay?=4 (a – 2a)3. -
25. Show that the locus of the extremity of the polar sub-
tangent of the curve
w=f(6),
is - u-- f' (;+)-0.
E. D. C. 8
114 DIFFERENTIAL CALOULUS,
26. Show that the locus of the extremity of the polar sub-
normal of the curve
r=f(6),
is r=f' (0-3).
27. In the curve
6 6
7" (m+ntan #)-1+tan 2 ”
show that the locus of the extremity of the polar subtangent is
m; 7?, r= 1 + cos 6.
CHAPTER IX.
ASYMPTOTES.
100. DEF. If a straight line cut a curve in two
points at an infinite distance from the origin and yet is
not itself wholly at infinity, it is called an asymptote to
the curve.
101. To obtain the Asymptotes.
If $ (q, y)=0..................... (1)
be the equation of any rational algebraic curve of the
n" degree, and
$/ = m100 + 0 . . . . . . . . . . . . . . . . . . . . . . (2)
that of any straight line, the equation
ºb (a, ma; + c) = 0 • e º e º O s e º 'º e º 'º - e o ºs º (3)
obtained by substituting the expression ma; + c for y
gives the abscissae of the points of intersection.
This equation is in general of the n” degree, showing
that a curve of the m” degree is in general cut in
m points real or imaginary by any straight line.
The two constants of the straight line, viz. m. and c,
are at our choice. We are to choose them so as to
make two of the roots of equation (3) infinite. We
then have a line cutting the given curve so that two
of the points of intersection are at an infinite distance
from the origin.
8—2
|16 JDIFFERENTIAL CA LOUI, U.S.
Imagine equation (3) expanded out and expressed in
descending powers of a; as
Aa" + Bajº 1 + Ca"-" -- ... + K = 0 ...... (4),
A, B, C, etc. being certain functions of m, and c.
The equation whose roots are the reciprocals of the
roots of this equation is
A + B2 + 02* + ... + K2" = 0
(by putting w = !) ;
2.
and it is evident that if A and B be both zero two roots
of this equation for 2 will become evanescent, and
therefore two roots of the equation for a become infinite.
If then we choose m and c to satisfy the equations
A = 0, B = 0,
and substitute their values in the equation
y = ma) + c,
we shall obtain the equation of an asymptote.
102. It will be found in examples (and it admits
of general proof) that the equation A = 0 contains m
only and in a degree not higher than m. Also that
B = 0 contains c in the first degree. Hence a curve of
the n” degree does not possess more than n asymptotes.
Ex. Find the asymptotes of the curve
g”—a:”y + 2y”--4y +a;= 0.
Putting y = ma;+ c,
(ma;+ c)” – arº (ma;+c) + 2 (ma;+ c)*-i- 4 (ma;+c)+a;=0,
Ol' (mº – m) a”--(3m”c – c +2m”) w”-- ... etc. =0.
We now are to choose m and c so that
m3 – m = 0
and 3m”c – c +2m2 =0ſ
The first equation is a cubic for m and gives mi-0, 1 or -1.
ASYMPTOTES. 117
The second equation is of the first degree in c and gives
C = 2m2
T 1 – 3m2
If m=0 we have c=0;
if m = 1 we have c = –1 ;
if m = - 1 we have c = − 1.
Hence we obtain three asymptotes, viz.
y =0,
g = a – 1,
y = – a – 1.
EXAMPLES.
Find the asymptotes of
1. 3/8–64:/*-i- 11a*) – 64%-- a--y=0.
ſy” – 44%) – wº/*-i- 4a:9-1-4ay — 44%–5.
2
3. 8ſ"–30°y+ay”–33:9-1-2/*-i-2ay+44'--5/4-6=0.
4. (y--a!-- 1) (y--2a-H2) (y-H 34-H 3) () – ar)+a++%–2=0.
5
(2a, +3)) (3a;+4y) (4a:--5y) +26a.” + 70+ y +47:/*-H2A-H 3/
= 1.
103. The case of parallel Asymptotes.
After having formed equation (4) of Art. 101
by substitution of ma, + c for y and rearrangement,
it sometimes happens that one or more of the values
of m, deduced from the equation A = 0, will make
B vanish identically, and therefore any value of c will
give a line cutting the curve in two points at infinity.
In this case as the letter c is still at our choice,
it may be chosen so as to make the third coefficient
C vanish. It will be seen from examples that each such
value of m now gives rise to two values of c. This is
the case of parallel asymptotes. The two lines thus
obtained each cut the curve at three points at infinity.
118 DIFFERENTIAL CALCULUS,
Ex. Find the asymptotes of the cubic curve,
!” – 5ay”--8a.”y — 4a:8–3y?--9ay – 649+ 2y – 2a:=1.
Putting ma;+ c for y and rearranging,
(m” – 5m” +8m – 4) a 9-1-(3m?c – 10mc--8c – 3m2+9m – 6) a.”
+(3mc” – 5c”–6mc +9e--2m –2) a + c2 – 3c2+2e – 1=0.
Choosing m3 – 5m” +8m – 4 = 0
and 3m?c – 10mc + 82 – 3m2+9m – 6=0ſ °
the first gives (m – 1) (m. – 2)*=0,
whence m-1, 2 or 2.
If m=1 the second equation gives c =0 and the corresponding
asymptote is y = w. -
If m=2 we have 12c — 20c 4-8c – 12+18 – 6 which wanishes identi-
cally for all finite values of c. Thus any line parallel to y = 2a: will
cut the curve in two points at infinity. We may however choose c so
that the next coefficient
3mc”— 5c” – 6mc +9C 4-2m – 2
vanishes for the value m = 2, giving
c” – 3c -i- 2 = 0, i.e. c = 1 or 2.
Thus each of the system of lines parallel to y = 2a: cuts the curve
in two points at infinity. But of all this infinite system of parallel
straight lines the two whose equations are
g = 2a: + 1,
and y = 2a:--2,
are the only ones which cut the curve in three points at infinity and
therefore the name asymptote is confined to them.
The asymptotes are therefore
Q) = &
Q) = 2a: + 1
EXAMPLES.
Find the asymptotes of
1. 3/8–ay”—aºy--a”--a”–3/*=1.
2. Aſ4–2ay3+2aºy–a4–34:84-3.cºy--3ay”—3/8–2a2+2%= 1.
3. (y”—aº)?–2 (~24-y”)=1.
ASYMPTOTES. 119
104. Those asymptotes which are parallel to the y-
axis will not be discovered by the above processes for
their equations are of the form a = a, and are not
included in the form y = ma + c for a finite value of m.
We, therefore, specially consider the case of those
asymptotes which may be parallel to one or other of
the co-ordinate axes.
105. Asymptotes Parallel to the Axes.
Let the equation of the curve be
a.a." + aya”y + aya”y” +...+ an—ay” + any”
+ bla^* + bºy + .................. + boy”
+ ca:"Tº + ...
+ ... = 0 ..................(1).
If arranged in descending powers of a this is
awa” + (any + bi) aſ” + ... = 0 ............ (2).
Hence, if a, vanish, and y be so chosen that
- any + b = 0,
the coefficients of the two highest powers of a in equa-
tion (2) vanish, and therefore two of its roots are infinite.
Hence the straight line any + b = 0 is an asymptote.
In the same way, if an = 0, an—a + b = 0 is an
asymptote.
Again, if a., - 0, a = 0, b = 0, and if y be so chosen
that
- (ty” + by + c2 = 0,
three roots of equation (2) become infinite, and the
lines represented by
asy” + by + c, - 0
represent a pair of asymptotes, real or imaginary, parallel
to the axis of a.
I 20 DIFFERENTIAL CA LOUI, U.S.
Hence the rule to find those asymptotes which are
parallel to the axes is, “equate to zero the coefficients of
the highest powers of a, and y.”
Ex. 1. Find the asymptotes of the curve
- a”y” — a y – a y”--a!-- y +1=0.
Here the coefficient of aº is y”— y and the coefficient of y” is a 2–a.
Hence ac-0, a = 1, y = 0, and y = 1 are asymptotes. Also, since the
curve is one of the fourth degree, we have thus obtained all the
asymptotes.
Ex. 2. Find the asymptotes of the cubic curve
w8+2a:”y +ay”—aº—ay--2=0.
Equating to zero the coefficient of y” we obtain a = 0, the only asym-
ptote parallel to either axis,
Putting ma;+ c for y,
a"--2a:” (ma;+c) + æ (ma;+c)” – a 3– a (ma;+c)+2=0,
or rearranging,
a" (1+2m+m”)+a” (20 +2me – 1 — m) + æ (c”—c)+2=0,
1+2m+m}=0 gives two roots m = — 1. 2C+2mc – 1 — me 0 is an
identity if m = –1 and this fails to find c. Proceeding to the next
coefficient c” – c = 0 gives c-0 or 1. -
Hence the three asymptotes are a = 0, and the pair of parallel lines
gy +a;=0,
g--a:= 1.
EXAMPLES.
1. The asymptotes of y” (a.”— a”)=w are
3/=0 }
a: = + aſ
2. The co-ordinate axes are the asymptotes of
a yº-Faºy=a+.
3. The asymptotes of the curve a yº–6° (cº-i-y”) are the
sides of a square.
106. The methods given above will obtain all linear
asymptotes. It is often more expeditious however to
ASYMPTOTES. 121
obtain the oblique asymptotes as an approximation of
the curve to a linear form at infinity as described in the
next article.
107. Form of the Curve at Infinity. Another
Method for Oblique Asymptotes.
Let P, F, be used to denote rational algebraical
expressions which contain terms of the r" and lower, but
of no higher degrees.
Suppose the equation of a curve of the m
be thrown into the form
(aa) + by + C) Pi—, 4-F, , = 0............ (1).
Then any straight line parallel to aw -- by = 0 obviously
cuts the curve in one point at infinity; and to find the
particular member of this family of parallel straight
lines which cuts the curve in a second point at infinity,
let us examine what is the ultimate linear form to
which the curve gradually approximates as we travel to
infinity in the above direction, thus obtaining the ulti-
mate direction of the curve and forming the equation
of the tangent at infinity. To do this we make the a
and y of the curve become large in the ratio given by
th
degree to
a : y = — b : a,
and we obtain the equation
aa. -- by + c + lºw----, (...) = 0.
If this limit be finite we have arrived at the equa-
tion of a straight line which at infinity represents the
limiting form of the curve, and which satisfies the
definition of an asymptote.
To obtain the value of the limit it is advantageous
to put a = –% and y = % , and then after simplification
make t = 0.
122 DIFFERENTIAL CALCULUS,
Ex. Find the asymptote of
a "4-3a*y--3ay°4-2y2=a^+y^+a.
We may write this curve as
(a + 2y)(a.”--acy + y”)=a^+ y” + ar,
whence the equation of the asymptote is given by
a:44-y”--a:
***u-It--,-, i.
and putting *-** ; :/ =} we have
4 + 1 2
3 T 3 T : 5 – 2t 5
a + 2y = Lt.-0 4 2 1 =Lt-o-º-;,
3 T 3 T 15
* s 5
7. 6., a + 2y = 3 *
EXAMPLE. Show that a +y =; is the only real asymptote of the
Curve (a +y) (24+y}) = a (c4+a+).
108. . Next, suppose the equation of a curve put
into the form
(aa) -- by + c) F. + Fn−2 = 0,
then the line aa, + by + c = 0 cuts the curve in two
points at infinity, for no terms of the n" or (n − 1)"
degrees remain in the equation determining the points
of intersection. Hence in general the line
aa. -- by + c = 0
is an asymptote. We say, in general, because if F, ,
be of the form (aa + by +c) Pi—s, itself containing a
factor aa. -- by + c, there will be a pair of asymptotes
parallel to aa, + by + c = 0, each cutting the curve in
three points at infinity. The equation of the curve
then becomes
(aa, + by + c)*P, a + Fn−s = 0,
ASYMPTOTES. - 123
and the equations of the parallel asymptotes are

amº-º-º-º-º-º-º-º-º
aa. --by-- e-V- Lt *— 5
77–2
where a and y in the limit on the right-hand side be-
b
come infinite in the ratio + = — a
Or, if the curve be written in the form
(aa) + by)*P, a -- (aa + by) R', a + f_2 = 0,
in proceeding to infinity in the direction aa + by = 0,
we have
Fn− r, ſn-
*T* -- Li "." Tº = 0
#4 Iº-0.
tº º e & b
when the limits are to be obtained by putting a = —
(aa -- by)* + (aa) + by). Lt
$/ = }, and then diminishing t indefinitely. We thus
obtain a pair of parallel asymptotes,
aa. --by = 0 and aa, + by = 8,
where 0 and 8 are the roots of
Fn−. Jº-
* - -- " - */ " ... —
And other particular forms which the equation of
the curve may assume can be treated similarly.
Ex. 1. To find the pair of parallel asymptotes of the curve
(2a – 3ſ +1)*(x+y)–8a;+2y –9–0.
Here 2a – 3y +1= Vºrº
a + y
where a, and y become infinite in the direction of the line 2a: =3/.
y
* 3 sº *
Putting a =; , v=} , the right side becomes +2. Hence the
asymptotes required are 2a – 3y = 1 and 2a – 3y +3=0.
124 DIFFERENTIAL CALCULUS,
Ex. 2. Find the asymptotes of
(a, -y)*(a.”+y”) – 10 (a, -y) w”--12y” +2a: +y=0.
2
Here (a, -y)*–10 (a, -y) Lt.-y-. zºº +12Lt.-, ºr 0.
Or (a – y)*– 5 (a — y) +6=0,
giving the parallel asymptotes a -y =2 and a — y=3.
109. Asymptotes by Inspection.
It is now clear that if the equation F. - 0 break up
into linear factors so as to represent a system of m
straight lines, no two of which are parallel, they will be
the asymptotes of any curve of the form
F., + Fn−2 = 0.
Ex. 1. (a – y) (a +y) (a + 2y – 1)=3a;+4y +5
is a cubic curve whose asymptotes are obviously
a – y = 0,
a + y = 0,
a + 2y – 1 = 0.
Ex. 2. (a — y)*(a + 2y – 1)=3a;+4y +5.
Here a + 2y – 1=0 is one asymptote. The other two asymptotes
are parallel to y=a. Their equations are

3 + 4 + 5t 7
*-y- “M Lt-o-Hig-i- *V.
110. Case in which all the Asymptotes pass
through the Origin.
If then, when the equation of a curve is arranged in
homogeneous sets of terms, as
wn + ºn–3 + ºn–3 + ... = 0,
it be found that there are no terms of degree n – 1, and
if also w, contain no repeated factor, the n straight lines
passing through the origin, and whose equation is un = 0,
are the n asymptotes,
ASYMPTOTES. I 25
EXAMPLES.
Find the asymptotes of the following curves:–
1. ºft-aº (2a–w). 2, 3/8–a, (a” – aſ”).
3. w8+y}=a^. 4. Q (a”--a”) = a”w.
5. aay =a^- a”. 6. /*(2a–w)=&º.
7. a 4-H /*=3aay. - 8. a7%/+% = 0.8.
9. Afty}=(a +y)*(b%–?). 10. wº–oº/*—b%2.
11. a y (w-)—a (a.”—y”)=b%. 12. (a”—a')?=a.” (a^+w”).
13. ay”=4a” (20–aj. 14, ſº (a – a =w (b-a)”.
15. a.”y=3:9-1-3,4-y. 16. ay”--a”y=a^+mo" + my 4-p.
17. cº-H2a”y—ay”–2/*-ī-4/*-ī-2ay+/– l =0.
18. aš–2aºy--ay?--a”—ay+2=0.
19. ºf (a, -y)*=y (a, -y)+2.
20. aš-i-2a:”y — 4a/*–8/°–40, H-8/= 1.
21. (a +y)*(x+2}/+2)= a +9/-2.
22. 33:94-17a.”y+21.0:y”–9/8—20.6%–12aay – 180/*–3a*w-i-a”y
111. Intersections of a Curve with its Asym-
ptotes.
If a curve of the n" degree have n asymptotes, no
two of which are parallel, we have seen in Art. 109 that
the equations of the asymptotes and of the curve may
be respectively written
F, - 0,
and F, -- F, , = 0.
The n asymptotes therefore intersect the curve again
at points lying upon the curve F., a = 0. Now each
asymptote cuts its curve in two points at infinity, and
therefore in m – 2 other points. Hence these n (n − 2)
points lie on a certain curve of degree m – 2. For
example,
126 DIFFERENTIAL CALOULUS,
1. The asymptotes of a cubic will cut the curve
again in three points lying in a straight line;
2. The asymptotes of a quartic curve will cut the
curve again in eight points lying on a comic
Section;
and so on with curves of higher degree.
EXAMPLES.
1. Find the equation of a cubic which has the same asym-
ptotes as the curve w8–6aºy--11ay”–6/8-|-a--y-1-1 =0, and which
touches the axis of y at the origin, and goes through the point
(3, 2).
2. Show that the asymptotes of the cubic
a!”y–ay”--ay-Hy”--a –y=0
cut the curve again in three points which lie on the line
a;+y=0.
3. Find the equation of the conic on which lie the eight
points of intersection of the quartic curve
ay (wº-y”)+a”y?--b%2–d?b”
with its asymptotes.
4. Show that the four asymptotes of the curve
(w”—y”)(y”–44%) –638-1-53%) +3ay”–298–39--3ay – 1 =0
cut the curve again in eight points which lie on a circle.
112. Polar co-ordinates.
When the equation of a curve is given in the form
ºf (6) +f, (6) = 0.................. (1),
it is clear that the directions given by
are those in which r becomes infinite.
Let this equation be solved, and let the roots be
a, S, Y, etc.
ASYMPTOTES. 127
Let xOP=a. Then the radius OP, the curve, and
ſhe asymptote meet at infinity towards P. Let Oy (= p)
P
O_2^ X
be the perpendicular upon the asymptote. Since OY
is at right angles to OP it is the polar subtangent, and
d6 22, Sr. r º
= — . . Let XOY = 2', and let Q be any point whose
p=-g
'll,
co-ordinates are r, 6 upon the asymptote. Then the
equation of the asymptote is
p = r cos (0-3)................... (3).
It is clear from the figure that a' = a – ; º
To find the value of d6 When u = 0, write .
du
for r in equation (1), and we have
f(0) + uſ, (0) = 0.
Whence differentiating
f' (6) + uſ, (6) + º f, (0) = 0.

128 DIFFERENTIAL CA LOUI, U.S.
Putting 6 = 0, and therefore u = 0, we have (if f'(a)
be finite
du w = 0 f'(a) & e a tº e º e º e s ∈ e º 'º e º e ºs g
Substitute this value of (— #) for p in equation
Q = 0
(3) and we obtain
0 (0. e
% = 7° COS (0– O. H. ..) = r sin (a – 6).
Hence the equations of the asymptotes are
_f. (3)
f'(a) '
r sin (8-0-%.
etc.
r sin (2-6)
113. Rule for Drawing the Asymptote.
After having found the value of (— º imagine
- = ()
we stand at the origin looking in the direction of that
value of 6 which makes u = 0. Draw a line at right
angles to that direction through the origin and of length
equal to the calculated value of (— #) to the right
u-0
or to the left, according as that value is positive or
negative. Through the end of this line draw a perpen-
dicular to it of indefinite length. This straight line
will be the asymptote.
Ex. Find the asymptotes of the curve
r cos 0– a sin 9=0.
PIere f (0)=cos 6 and f, (9)= – a sin 6,
3
cos 0 = 0 gives a=;, 8 = ..., etc.
ASYMPTOTES. 129
and the length of the polar subtangent
fo(a) – a sin a
Tjºſa) T ~sina T
Hence the equations of the asymptotes are
... / Tr . ... / 37ſ
7° Sln (;- b)=a and r sin (; - 0)=a,
i.e. r cos 0 = a and r cos 0 = - a.
These are perpendicular to the initial line and at distances a respec-
tively to right and left of the origin.
EXAMPLES.
Find the asymptotes and draw their positions for the follow-
ing curves:–
1. ré–a. 2. ré) = a.
3. * sin 7.6=a. 4. r = a cosec 6-H b.
5, r =2a sin 6 tan 6. 6. r sin 26= a cos 36.
7. rea-i-b cot nó. 8. 7” sin mé– a ".
114. Circular Asymptotes.
In many polar curves when 6 is increased indefinitely
it happens that the equation ultimately takes the form
of an equation in r which represents one or more con-
centric circles.
ge 6
IFor example, in the curve r = a GTI'
which may be written 7° E (I,
it is clear that if 6 becomes very large the curve approaches in-
definitely near the limiting circle r- a.
Such a circle is called an asymptotic circle of the
CUITVé.
E. D. C. 9
130 DIFFERENTIAL CALCULUS,
EXAMPLES.
Find the asymptotes of the following curves:—
1, wº–y'-a'ay. 2, wº–yº–a4,
2
3. aft–yº = agſ". 4. y–ºt e
º —a ſº-tºa-º F (a, -3)
5. (a – a) /*=(2a – vya". 6. / (~T)(x-2)
y” tº — r\4— ox! — or'ſ
7. gy–1 w–2 8. (y – wyº-wº—a".
9. (3:4–y”)?=a^*—bºy”. 10. y” (a”—vº)=(wº–2a3)”.
y” (a — a.)”
11. º– cº-Ea? " 12. $/8-(a – 1)*(a — 4).
13. (y—w)” wº—4 (y-a)*--16. 14. (wº-4%)}=2 (v34-y”).
15. ay” — y=aw”--bº” + cz + d.
16. (y–23)” (3.0-4-4ſ)+3 (y–23) (304-43)=5.
17. (y–2a)*(344-44)--3 (y–20)(344-4/)+11/=5.
18. yº–4ay°4-30%+44'y–4a:
+3/8–6ay”–3a”y +63:34-2/?–20% =a.
19. "- b sec a 6. -
20. r("–1)=a ("+1).
Find also the circular asymptote.
21, r (6%–1)=a&”.
Find also the circular asymptote.
" sin 6 e
22. 7 COS 6 = 2a: I-Hsin 6 23. 7' cos 26= a sin 36.
24. Show that the asymptotes of
(w?–?)?=2 (v*-ī-y”)
form a square.
25. Show that the asymptotes of
a’y” – a” (cº-Fy”)—aº (c-i-y)+a+=0 -
form a square, through two of whose angular points the curve
passes.
EXAMPLES. 131
26. Show that the asymptotes of
a?--24%/–ay”–2/3+3%–?=204-3/
cut the curve again in three points collinear with the origin.
27. Show that the asymptotes of the quartic
a"—5a”y”--4/4+3°–ay?–2aºy-H2/8—ay=1
cut the curve again in eight points lying upon a rectangular
hyperbola.
28. Show that the asymptotes of the quartic
(wº-y”)(39–4/*)+20%–20%+20%–2ay=2~ +2/– I
cut the curve again in eight points lying upon a parabola which
touches the co-ordinate axes.
29. Show that the three quartics
(a) a y (wº-y”)+2a4+y}=1,
(b) ay (a"—y”)+a;%+2/*= 1,
(c) aſy (a!”—y”)+2a2+2ay+2/*= 1
have the same asymptotes; and that each of the three conics
on which lie the other eight points of intersection of each curve
with its asymptotes have double contact with a certain circle.
30. Find the asymptotes of the sextic
w/ (wº-y”) (a’—4/*)+2aºy (wº–y”)+5.4°4-5/*=1.
Show that they cut the curve again in twenty-four points
lying upon a certain quartic. Find the equation to this quartic
and show that its own asymptotes are common with four asymp-
totes of the sextic and that they cut the quartic again in eight
points lying upon a circle. Also that the remaining asymptotes
of the sextic are tangents to this circle.
CHAPTER X.
CURVATURE.
115. Angle of Contingence.
Let PQ be an arc of a curve. Suppose that between
P and Q the bending is continuously in one direction.
Let LPR and MQ be the tangents at P and Q, inter-
Z
secting at T and cutting a given fixed straight line LZ
in L and M. Then the angle RTQ is called the angle
of contingence of the arc PQ.
The angle of contingence of any arc is therefore the
difference of the angles which the tangents at its ex-
tremities make with any given fixed straight line. It
is also obviously the angle turned through by a line
which rolls along the curve from one extremity of the
arc to the other.

CURVATURE. 133
116. Measure of Curvature.
It is clear that the whole bending or curvature which
the curve undergoes between P and Q is greater or less
according as the angle of contingence RTQ is greater or
angle of contingence :
is called the
length of arc
average bending or average curvature of the arc. We
shall define the curvature of a curve in the immediate
neighbourhood of a given point to be the rate of deflection
from the tangent at that point. And we shall take as
a measure of this rate of deflection at the given point
angle of contingence when
length of arc
the length of the arc measured from the given point,
and therefore also the angle of contingence are inde-
finitely diminished.
less. The fraction
the limit of the expression
117. Curvature of a Circle.
In the case of the circle the curvature is the same at
every point and is measured by the RECIPROCAL OF THE
RADIUS, -
For let tº be the radius, O the centre. Then
R70– P60 -** 5
the angle being supposed measured in circular measure.

134 DIFFERENTIAI, OAJLOUI, U.S.
Hence
angle of contingence 1
length of arc r
j
and this is true whether the limit be taken or not.
Hence the “curvature” of a circle at any point is
ºmeasured by the reciprocal of the radius.
118. Circle of Curvature.
If three contiguous points P, Q, R be taken on a
curve, a circle may be drawn to pass through them.
When the points are indefinitely close together, PQ and
QR are ultimately tangents both to the curve and to
the circle. Hence at the point of ultimate coincidence
the curve and the circle have the same angle of con-
tingence, viz. the angle RQZ (see Fig.). Moreover,
the arcs PR of the circle and the curve differ by a
small quantity of order higher than their own, and
therefore may be considered equal in the limit (see Art.
81). Hence the curvatures of this circle and of the
curve at the point of contact are equal. It is therefore
convenient to describe the curvature of a curve at a
given point by reference to a circle thus drawn, the

CURVATURE. 135
reciprocal of the radius being a correct measure of the
rate of bend. We shall therefore consider such a circle
to exist for each point of a curve and shall speak of it
as the circle of curvature of that point. Its radius,
and centre will be called the radius and centre of
curvature respectively, and a chord of this circle
drawn through the point of contact in any direction
will be referred to as the chord of curvature in that
direction.
119. Formula for Radius of Curvature.
Let PQ and QR be considered equal chords, and
therefore when we proceed to the limit the elementary
arcs PQ and QR may be considered equal. Call each
8s, and the angle RQZ = 64!.
Now the radius of the circum-circle of the triangle
PQR is
PR
2 sin PQR
Hence if p be the radius of curvature, we have
– L = ** – Il-º-
p 2 sin PQR T "" 2 sin 8,
r, 88 8-| ds
* *; sin & T dy. ........(A).
Also, it is clear that the lines which bisect at right
angles the chords PQ, QR intersect at the circum-
centre of PQR, i.e. in the limit the centre of curvature
of any point on a curve may be considered as the point
of intersection of the normal at that point with the normal
at a contiguous and ultimately coincident point.
120. The formula (A) is useful in the case in which the equation
of the curve is given in its intrinsic form, i.e. when the equation is
given as a relation between s and p. For example, that relation for
a catenary is 8 = c tan p, whence
p=w= secºp,
136 DIFFERENTIAL CA LOUI, U.S.
and the rate of its deflection at any point is measured by
1 cos’ſ c
pT e Tsº-c”
121. Transformations.
This formula must be transformed so as to suit each
of the systems of co-ordinates in which it is usual to
express the equation of a curve. These transformations
we proceed to perform. :
We have the equations
*- da, ſº *ms dy
cos W =;, sin k = i.
Hence, differentiating each of these with respect to s,
3, ... dº dº . dy dºy
-sin biº-i. cos / º – i.
- _dºw dy
I ds, dsº
whence p T + *- T tº e g º e º tº º ſº e º 'º º e º ſº e º 'º º (B),
ds ds
and by squaring and adding
I *mº dža)? dºy 2
These formulae (B) and (c) are only suitable for the
case in which both a and y are known functions of S.
122. Cartesian Formula. Explicit Functions.
Again, since tan ºr = % 5
... dº dy
we have secº ºr da, T day. '
by differentiating with regard to a.
O'UR VATURE. 137
dal diff ds 1 .
Now de Tds de To cosº,
1 dºy
* Aº ‘ " Arººre 3 — = —-
therefore secº le. da” ”
dy)*
and secº || = 1 + tan” ) = 1 + (%) ;
2) #
{1+(#)}
therefore p = + —H-.… (D).
da,”
This important form of the result is adapted to the
evaluation of the radius of curvature when the equation
of the curve is given in Cartesian co-ordinates, y being
an explicit function of ac.
Ex. In the curve !y = log sin a',
we have !/1=cot ºc,
Aſsis — cosec”a.
3
2\:
p = + (1+y,”)?
!/2
_(1+cot? a)#
T cosec”a:
Hence
- COSOC (0.
123. Curvature at the Origin.
When the curve passes through the origin the values
of % (= p) and º (= q) at the origin may be deduced
without actual differentiation by substituting for y the
expression pa -- º + ... (the expansion of y by Mac-
laurin's Theorem) and equating coefficients of like
138 DIFFERENTIAL CALCULUS,
powers of a, in the identity obtained. The radius of
curvature at the origin may then be at once deduced
from the formula
2\}
p = + º [Formula (D)].
Ex. Find the curvature of the conic y – a = w”--2ay + y” at the
origin.
g ac?
Putting y=pºt agſ--...
qa’, .2 22.2
we have (p-1)***, +... =w +2pa!”--pºw”--...
identically;
whence by equating coefficients of like powers of a.,
p=1, q = 2 (p + 1)*=8,
3
2\º g
(***-2-3-3535.
and F.
p Q
124. If we apply the same method to the general curve
aa. -- by
+a'a.” +2h'ay +b'y”
+ ...... =0,.................... (1),
we obtain after substituting
qa.” , *
pºt ºf + ... for y,
and collecting the powers of a.,
(a+b)+(******)2+...+0.
Hence a +bp=0,................................ (2),
, , or,. . . . . "4
a'+2hp#Wºº-Hº-0 * * * * * * * * * * * * e º e º 'º º e º e º 'º e (3),
etc.,
CUR VATURE. 139
(l, a'+2h'p +b'p”
giving p=-j. q = -2 b , etc.,
3 3.
(1+p”); 1 (a2+b^);
= + -t- = — --→- -- -- -
whence p Q 2 aſb2 – 2k'ab + b/a2 ‘’’ ‘’’ ‘’’ ‘’’ ‘’’ (4),
the value of the radius of curvature of the given curve at the origin.
125. Tangents at the Origin. Double point.
It will be noted that the equation
9––*
p=-
indicating the value of # at the origin proves the equation of the
tangent there to be
y
(l,
----- y- Y=
A. b ' or a Y + b Y= 0,
which therefore might be at once written down as being the terms of
the lowest degree in the equation of the curve (see Art. 87).
When no linear terms occur in the equation of the curve we have
a = 0 and b = 0, and the value of %
mined form. We however obtain from equation (3) the quadratic
at the origin takes an undeter-
b'p”--2h'p + a' =0 ........................... (5),
giving two values of p at the origin. It is thus indicated that in this
case two branches of the curve pass through the origin in the direc-
tions given by equation (5). The tangents to the curve at the origin
are therefore
a'X?--2h'XY-H b'Y2–0,
a result which may be written down by inspection from the equation
of the curve, by equating to Zero the second degree terms; i.e. the
terms of lowest degree (Art. 87). The origin is now said to be a
double point upon the curve.
The curvatures of the two branches at the double point may be
obtained by the same method as before (Art. 123) as shown in the
following example.
Ex. Find the radii of curvature at the origin for the curve
y°–3ay +2a:” – a "+y}=0,
140 DIFFERENTIAL CALOUI, U.S.
q
Substituting pa: +jº: ... for y, and collecting the powers of a.,
we have (*-*****(n+1-1)+...+0.
whence p°–3p +2=0,
- 3
pa-34–1=0,
etc.,
whence p = 1 or 2,
and q = -2 or 2,
and therefore - (1 +p°)} - 2} = — V2= – 1414
p Q — 2 T * • * > y
# 5 5 = 5'590
Or = 2 = 5^/ -: *) * :) e e s º
The difference of sign introduced by the q indicates that the two
branches passing through the origin bend in opposite directions.
R

\
`s N – X
A
126. Newtonian Method.
The Newtonian Method of finding the curvature of
the curve at the origin is instructive and interesting.
Suppose the axes taken so that the axis of a is a tangent
to the curve at the point A, and the axis of y, viz. AB,
CUR VATURE. 141
is therefore the normal. Let APB be the circle of
curvature, P the point adjacent to and ultimately co-
incident with A in which the curve and the circle inter-
sect; PN a perpendicular upon AB. Then
PNP = AN. NB,
PN2
OI NB = ºr,
Now in the limit
NB = AB = twice the radius of curvature.
I PN2 ag?
‘ā’īy-ſº
Similarly, if the axis of y be the tangent at the
Hence p = L
© tº y”
origin, we have p = Lt 2n'
Ex. Find the radius of curvature at the origin for the curve
2.c4+3/4+4a:”y +ay — y” +2a:=0.
In this case the aaris of y is a tangent at the origin, and therefore
2
we shall endeavour to find Lt. e
2 a 12
Dividing by a , 23:34-37%. ": +4ay + y – º + 2 = 0.
Now, at the origin Lt ". ==2p, a = 0, y = 0, and the equation becomes
–2p +2=0,
OI’ p=1.
EXAMPLES.
1. Apply formula (A) to the curves
s = avy, 8 = a sin Jr, S = a secº ſº,
- T , ,
s= a log tan (; + #) º
I42 DIFFERENTIAL CALCULUS,
2. Apply formula (D) to the curves
3/=a", § – c cosh g’ v=alog see .
3. Apply formulae (B and C) to the curve for which
S
** = C, CO.S —
C!,
si S
'?/ = C, SII] —
3/ CM,
4. Prove that in the case of the equiangular spiral whose
intrinsic equation is
s= a (e" – 1),
p=mae”.
5. For the tractrix s = c log sec / prove that
p = c tan Jr.
6. Show that in the curve
3/= a +33*—aº
the radius of curvature at the origin = 4714, .. , and that at
the point (1, 3) it is infinite.
7. Show that in the curve
y”–3ay – 44%--a”--aºy--yº–0
the radii of curvature at the origin are
º V17 and 5 V2.
8. Show that the radii of curvature of the curve
Q. -- 0:
2 = 0:2
3/ C!, - ſº
for the origin = + a V2,
and for the point (– a, 0) =} &
9. Show that the radii of curvature at the origin for the
CUlrWe
w8+y}=3avy
_3a.
are each 2
CU/º VA TURE. 143
10. Prove that the chord of curvature parallel to the axis
of y for the curve
= a log Sec w
y=a logsee;
is of constant length.
11. Prove that for the curve
s=m (secº || – 1),
p=3m tan / Secº ſº,
and hence that
Also, that this differential equation is satisfied by the semicubical
parabola
27my?=8w8.
12. Prove that for the curve
s= a log cot (- %) + (l, sin J.
2 cos”, ”
p=2a secº y :
and hence that
dº. 1
dw? 20 °
and that this differential equation is satisfied by the parabola
a’=4ay.
13. Show that for the curve in which s–ae"
cp=s (8*— 62)}.
14. Show that the curve for which s = V8oy (the cycloid)
has for its intrinsic equation
s= 4a sin Jr.
Hence prove p = 4a. w/ 1 — 3. ©
144 DIFFERENTIAL CA LOULUS,
15. Prove that the curve for which y”=cº--s” (the catenary)
has for its intrinsic equation
8=ctan J.
y”
c.
tween the curve and the az-axis.
Hence prove p =%--the part of the normal intercepted be-
127. Formula for Pedal Equations.
Since a curve and its circle of curvature at any point
Pintersect in three contiguous and ultimately coincident
points they may be regarded as having two contiguous
tangents common. Therefore the values of r + 8r and
p + 8p are common in addition to those of , and p; i.e.
the value of º is common. Now let O be the pole and
C the centre of curvature corresponding to the point P
on the curve.
Then OC* = r + p” – 2rp cos OPC
=r" + p^-2rp sin (b
= r. pº-2pp.

OUR WATURE. 145
Considering this as referring to the circle (for which 00
and p are constant) we obtain by differentiating
dr
() = 2r dp rººmsº 2p,
and it has been pointed out that the values of r and
º, are the same at the point P for the curve and for
the circle. Hence for the curve itself we also have
Ex. In the equation p”= A^* + B, which represents any epi- or
hypocycloid [p. 113, Ex. 23], we have
d?"
p=A7 dp'
and therefore p OC p.
The equiangular spiral, in which p oc r, is included as the case in
which B=0.
128. Polar Curves.
We shall next reduce the formula to a shape suited
for application to curves given by their polar equations.
We proved in Art. 95 that
1 ..., , /du)*
:=wt ( ) ©
p” .
d6
Now p=. and r=}
146 DIFFERENTIAL CA LOUI, U.S.
therefore - - }; - —- t
w" ap pºu? (u + d º
092
Qı” + º
(i.
Ol' . . p=–F– e to $ tº e < * * * * * * e º e º 'º we º e e (G).
Qış (u + #)
- d62/.

129. This may easily be put in the r, 6 form
thus:—
Since QI, E I 5
7"
we have - du * =ss 1. dr
d6 rº d6'
dºw 2 /dry? I dºr
and therefore d63 - r; (%) * 7.3 d62 3.

therefore p = 7
130. Tangential Polar Form.
Let the tangent P T make an angle aſ with the
initial line. Then the perpendicular makes an angle
a = | – with the same line. Let 0Y = p. Let PP,
be the normal, and P, its point of intersection with the
normal at the contiguous point Q. Let 0Y, be the
perpendicular from 0 upon the normal. Call this pl.
CUR WATURE. I47
Let P.P., be drawn at right angles to P, P, and let the
length of 0Y, the perpendicular upon it from 0, be ps.
The equation of PT is clearly
p = a cos a + y sin a.................. (1).
The contiguous tangent at Q has for its equation
p + 8p = a cos (a + 82) + y sin (a + 82)...(2).
Hence subtracting and proceeding to the limit it ap-
pears that
}=-win a tycosa e is is e º e º º ºs e a e º e e (3)
is a straight line passing through the point of inter-
section of (1) and (2); also being perpendicular to (1) it
is the equation of the normal P. P.
tº gº dºp e
Similarly ſº--woosa-ysin a............... (4)
represents a straight line through the point of inter-
section of two contiguous positions of the line P, P, and
perpendicular to PP, viz. the line P, P, and so on for
further differentiations.

10—2
I48 DIFFERENTIAL CA LOUI, U.S.
From this it is obvious that
dp_ dp & dy 1.
0)-;=#. Sl]]CG i = 1;
d'p dºp
7. He ºf: - ºf-
OY =; dy”
etc.
o v_ "p.
Hence P, Y = day. ' -
r dºp ,
and p = P, P = 0Y + 0Y, - p + .....(1).
dil,
This formula is suitable for the case in which p is given
in terms of Je.
Ex. It is known that the general p, p equation of all epi- and
hypocycloids can be written in the form
p = A sin Bºp.
Hence p=A sin By – AB” sin Bºp,
and therefore p Oc p.
131. Point of Inflexion.
If at some point upon a curve the tangent, after its
cross and recross, crosses the curve again at a third
ultimately coincident point, as shewn magnified in the

figure, the point is called a point of inflexion. At such
a point the two successive chords PQ, QR are in line
and the angle of contingence vanishes.
CUR VATURE. 149
At a point of inflexion the circle of curvature passes
through three collinear points, and the radius of curva-
ture becomes infinite and changes sign. We may hence
deduce various forms of the condition for a point of
inflexion; thus if
WG get
2 dr
p = CO,
% = 0 from (A),
%– 0 from (D),
20 + º = 0 from (G),
) — ?” * = 0 from (H).
132. List of Formulae.
The formulae proved above are now collected for
convenience.
ds
p F. dy is e º e º e º e º e s a e s s e º s e e º e e º e º e º 'º e º e e (A),
dºw dºy
I ds” ds”
p ---, -º 77 da, e e º 'º e e s = n e º e s e < * * * * * * * * (B),
ds ds
1 (da)” º)
p” ( ...) + (i. e e s & e º s 2 & º sº e < * * * * * (C),
# -
1 + y,”
p = dº is g º e º e º ºs g º e º n - e º 'º t e º 'º e g º º & (D),
a;2
p = Lt 2) & (E),
dr
O = ?' j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(F),
150 DIFFE/RENTIAL CA LOUI, US.
9 983
(u” +w,”)
* w”(u + tle) g g g tº s º ºs e e º e º & e g º ºs e a tº e º g e (G),
#
p =s**E=& 12 + 2r.” <-ºmº 7'r, * * * * * * * *s a º is tº e º 'º e º s (H),
d?
p-p+..… e e º g º ºs e º 'º e º 'º º 'º e º 'º e º e º ºs (I).
EXAMPLES.
1. Apply formula (F) to the curves
...??? -- 1
p”- ar, ap=7%, p=. e
2. Apply formulae (G), (H) to the curves
aw = 6, r = a&, r = a sin 6.
3. Apply the polar formula for radius of curvature to show
that the radius of the circle -
*= a cos 6 is
4. Show that for the cardioide r = a (1 +cos 6)
- –* cos”, .e., & Jr
p= 3 2 ; 2.6.3 ^ 1 -
Also deduce the same result from the pedal equation of the
curve, VIZ.,
P W2a=r8.
5. Show that at the points in which the Archimedean spiral
r= a& intersects the reciprocal spiral rô= a, their curvatures are
in the ratio 3 : 1.
6. For the equiangular spiral r- ae” prove that the centre
of curvature is at the point where the perpendicular to the
radius vector through the pole intersects the normal.
7. Prove that for the curve
* = a sec 26,
4.4
p = - 303 e
cuſ: VATURE.
15]
8. For any curve prove the formula
7°

p -:
sin q. ( -H %)
rdó
where tan q = or "
Deduce the ordinary formula in terms of r and 6.
9. Show that the chord of curvature through the pole for
the curve
p=f(r)
tº º dr ºf (r)
is given b chord =2p + = 2*, *{.
given by l *d, *f;()
10. Show that the chord of curvature through the pole of
the cardioide
r= a (1 + cos 6) is 3 *
11. Show that the chord of curvature through the pole of
the equiangular spiral
* = ae”9 is 27".
12. Show that the chord of curvature through the pole of
7” = a "cos mé is - T - .
7m + 1
I
the curve
Examine the cases when m = -2, — 1,
13. Show that the radius of curvature of the curve
* = a sin m6
700,
at the origin i — .
sº S 2
7" = aſh cosmó,
Oſm
P- (m-II) Fi
14. For the curve
Examine the particular cases of a rectangular hyperbola,
prove that
lemniscate, parabola, cardioide, straight line, circle.
152 JDIFFERENTIAL CA LOUI, U.S.
133. Centre of Curvature.
The Cartesian co-ordinates of the centre of curva-
ture may be found thus: —
Let Q be the centre of curvature corresponding to
the point P of the curve. Let 0X be the axis of a ;
0 the origin; aſ, y the co-ordinates of P; aſ, j those of

\
R
—T
º
O T M N X
P
Q; ºr the angle the tangent makes with the axis of 4.
Draw PN, QM perpendiculars upon the w-axis and PR
a perpendicular upon QM. Then
£ = OM = ON – RP
= ON – QP sin ºr
=w – p sin ºr,
and y = MQ = NP + RQ
= y + p cos le.
Now tan \le = };
therefore sin Aſ = Jº
and cos ºr = I
Ji-Hyº
CURVATURE. 153
3.
I 9Y2
Also p_0+ ft).
3/2
-------- I 2 &
Q} = {\} – !h (1 + yº) e e º e º is a e s ∈ a s = e tº (d),
Hence I 3/2
2
y=y+*** |… (8).
2
EXAMPLES.
1. For the parabola
y=a’ſ 4a,
a;= — wº 7 = 20. 34°
prove 46.2 ° y = 24-H 44
2. For the parabola
3/*=4aa,
prove a;=2a+3a, § = – 20*a*, p = 2SP3 ſa",
SP being the focal distance of the point of the parabola whose
coordinates are (v, V).
3. For the ellipse
w”/a2+y}|b2–1,
a”- 0°, a 3, b%– a”, a2b%
* @9, 7/ = --—H-, 2/" - —-
a 4 ; 3/ b% 3/", p p3 5
prove a;=
p being the central perpendicular upon the tangent at (a, y).
4. For the cubical parabola
I’OWe 5–? 1_9. 7–" ** *
p “T2 of ), "T3 …?" 6,
CONTACT.
134. Consider the point Pat which two curves cut.
It is clear that in general each has its own tangent at
that point, and that if the curves be of the mº and m"
154 JOIFF/E/EENTIAL CA LOUI, U.S.
degrees respectively, they will cut in mm – 1 other
points real or imaginary.
Q
sº
*
*
Next, suppose one of these other points (say Q) to
move along one of the curves up to coincidence with P.
The curves now cut in two ultimately coincident points
at P, and therefore have a common tangent. There is
then said to be contact of the first order. It will be
observed that at such a point the curves do not on the
whole cross each other.
Again, suppose another of the mn points of inter-
Section (viz. R) to follow Q along one of the curves to
coincidence with P. There are now three contiguous
points on each curve common, and therefore the curves
have two contiguous tangents common, namely, the
ultimate position of the chord PQ and the ultimate
position of the chord Q.R. Contact of this kind is said
to be of the second order, and the curves on the whole
cross each other.
Finally, if other points of intersection follow Q and
R up to P, so that ultimately k points of intersection



CUE WATURE. 155
coincide at P, there will be k – 1 contiguous common
tangents at P, and the contact is said to be of the
(k-1)" order. And if k be odd and the contact there-
fore of an even order the curves will cross, but if k be
even and the contact therefore of an odd order they will
not CrOSS.
135. Closest Degree of Contact of the Conic Sections with a
Curve. -
The simplest curve which can be drawn so as to pass
through two given points is a straight line.
do. three do. circle.
do. four do. parabola.
do. five do. conic.
Hence, if the points be contiguous and ultimately coincident points
on a given curve, we can have respectively the
Straight Line of Closest Contact (or tangent), having contact of the
first order and cutting the curve in two ultimately coincident
points, and therefore not in general crossing its curve; the
Circle of Closest Contact, having contact of the 3econd order and cut-
ting the curve in three ultimately coincident points, and therefore
in general crossing its curve (this is the circle already investigated
as the circle of curvature); the
Parabola of Closest Contact, having contact of the third order and
cutting the curve in four ultimately coincident points, and there-
fore in general not crossing; and the
Conic of Closest Contact, having contact of the fourth order and cut-
ting the curve in five ultimately coincident points, and therefore
in general crossing.
It is often necessary to qualify such propositions as these by the
words in general. Consider for instance the “circle of closest contact”
at a given point on a conic Section. A circle and a conic section
intersect in four points, real or imaginary, and since three of these
are real and coincident, the circle of closest contact cuts the curve
again in some one real fourth point. But it may happen, as in the case
in which the three ultimately coincident points are at an end of one of
the axes of the conic that the fourth point is coincident with the other
three, in which case the circle of closest contact has a contact of
higher order than usual, viz., of the third order, cutting the curve
in four ultimately coincident points, and therefore on the whole not
crossing the curve. The student should draw for himself figures of
the circle of closest contact at various points of a conic section,
remembering that the common chord of the circle and conic and the
tangent at the point of contact make equal angles with either axis.
The conic which has the closest possible contact is said to osculate
156 DIFFERENTIAL CALCUL U.S.
its curve at the point of contact, and is called the osculating comic. .
Thus the circle of curvature is called the osculating circle, the para-
bola of closest contact is called the osculating parabola, and so on.
136. Analytical Conditions for Contact of a
given order.
We may treat this subject analytically as follows.
Let y = b (a)]
$/ = h (a))
be the equations of two curves which cut at the point
P(a, y).
Consider the values of the respective ordinates at
the points Pi, P, whose common abscissa is a + h.
Let MIN = h.
Then NP = b (a + h),
NP, - ) (a + h),
and P, P = NP = NP, - b (a + h) — ſº (a + h)
= [q, (a) — J. (a)] + h [ºp' (a) – jº' (a)]
+:[*@-vºlt.
y-b (*)
wº
P.
Q M N X
If the expression for P. P. be equated to zero, the
roots of the resulting equation for h will determine the
points at which the curves cut.

OUR PATURE. 157
If p (a) = | (a), the equation has one root zero and
the curves cut at P.
If also p'(a) = \lº'(a) for the same value of a, the
equation has two roots zero and the curves cut
in two contiguous points at P, and therefore
have a common tangent. The contact is now of
the first order.
If also p" (a) = \!" (w) for the same value of a, the
equation for h has three roots zero and the
curves cut in three ultimately coincident points
at P. There are now two contiguous tangents
common, and the contact is said to be of the
Second order; and so on.
Similarly for curves given by their polar equations,
if r = f(0), r = q.(6) be the two equations, there will be
m + 1 equations to be satisfied for the same value of 6
in order that for that value there may be contact of the
m" order, viz.
f(0)= }(0), f(0) = } (0), f" (0) = $" (0), ......
f” (0) = dº (6).
EXAMPLES.
1. Shew that the parabola whose axis is parallel to the y axis
and which has the closest contact possible with the curve y=a^*
at the point (1, 1) is
3/=3–84; +64%.
2. Draw carefully the circle of curvature
(1) at an ordinary point on an ellipse,
(2) at the end of the major axis,
(3) at the end of the minor axis,
(4) at an ordinary point on a parabola,
(5) at the vertex of the parabola,
and name the order of contact in each case.
158 DIFFERENTIAL CA LOUI, U.S.
3. Shew that the curves
ſy=a^**(c-a)(e–b),
Ay=a^**(v-c)(3)-d),
have contact of the nº order at the origin.
EXAMPLES.
1. Find the curvature at the origin in the curves
(a) y=2a:--3a*-i-4g",
(b) y=2~ +3a*4-4ay,
(c) (y-a) (y–2a)=a^+y",
(d) (w-y)*(v–2/)(a -3%)=2a (a"—y”)+2a”(a +/)(e– 2/).
3:
2. In the curve 3/=ae",
* º
prove p = a sec” 6 cosec 6, where 6=tan-" 3/ }
(1.
and that the curve has no point of inflexion.
3. In the limaçon r = a + b cos 6,
_(a4+2abcos 6+ bºy;
"Taz-E3abcos & E252
and hence shew that if a. and b are both positive the limaçon can
only have points of inflexion when a is intermediate between b
and 2b.
Deduce for a cardioide (b-a),
prove
– a cos
p=3 a cos;
4. Shew that the curve y=e-* has points of inflexion where
w- +1/V2.
5. Shew that y=a sin . has points of inflexion wherever it
cuts the ac-axis.
6. Shew that the points of inflexion upon
a’y=a^ (a –?),
are given by a =0 and a = + a V3.
EXAMPLE'S. 159
7. Shew that the curve
a (a.”—ay)=a?
has a point of inflexion where it cuts the a-axis. Find the
equation of the tangent there.
8. Shew that the curve
a 34-y” – a”
has inflexions where it cuts the coordinate axes.
9. Shew that the curve a = log () has a point of inflexion
at (–2, —2e *).
10. Shew that r M6= a has a point of inflexion at a distance
a V2 from the pole. -
11. Find the points of inflexion upon
12/=a:4–1648-1-42a2+12a:4-1.
12. Shew that in a parabola the chord of curvature through
the focus and the chord of curvature parallel to the axis are each
four times the focal distance of the point of contact.
13. In a parabola the common chord of the curve and its
circle of curvature=8 Wr (r-a).
14. In the chainette y =c cosh: the chord of curvature
parallel to the y-axis is double of the ordinate, and that parallel
to the ac-axis
º 24:
=o sinh .
15. If C. and C, be the chords of curvature parallel to the
axes at any point of the curve
3/= ae",
prove that
1 1 1
Gºt GT3ad.
16. If C, and Co be the chords of curvature respectively
along and perpendicular to the radius vector, shew that
* — on?
c. *, c. ** p".
160 DIFFERENTIAL CALOULUS,
17. At the point upon the Archimedean Spiral
- *= a&,
at which the tangent makes half a right angle with the radius
vector,
4
prove C. = C6 = 5%
18. Shew that for each of the curves
r= aff, rô= a, rsin nô= a, rsinh né=a,
r cosh m6 = a (Cotes's Spirals),
p3
the curvature oc Aſ
19. Shew that in the curve for which
3/=a cos” ºr,
the radius of curvature is m times the normal.
20. Shew that in the curve for which
3/=ae”,
the radius of curvature is m times the tangent.
21. Shew that in the curve for which
72 – 0.2 + b%),
the radius of curvature varies inversely as the perpendicular
from the origin upon the normal.
22. Shew that in the chainette the radius of curvature varies
as the square of its projection upon the y-axis.
23. In the curve
»2–22
6 r Wrº-a” — COST1 a. 2
Cſ, 7"
shew that p°=2as,
s being so measured that p and 8 vanish together.
24. Prove that in any curve
d 9
# = {3/18/34–3/s (1+3/1%)}ly,”,
and shew that at every point of a circle
3/3+3/18/2”/(1 +8/1%).
I9T 'STTCIWWYT
II 'O "Oſ ºf
g8A2. #82 382. 1879
*—º-º-º-d (a)
S/2 . E8A2 L 8/2 . E8/2
ſºp ſtºp ap app
ºf " £2 º' gº º (o)
do gº tºp hºp Tºp ºr "
, Sp gd sp #80 S/2. Sp (9)
dp & T ſp ſºp" ap alſo
(;)+(...)={(i)+1};
+ * ---
(; z \@g? g\dp + I I (q)
ºf gº-gº re-º o
a'ºp ſtºp ſtºp aºp I
eAOId aaano augſd Kuſe IOI ga.
CHAPTER XI,
ENVELOPES.
137. Families of Curves.
If in the equation p(a), y, c) = 0 we give any arbitrary
numerical values to the constant c, we obtain a number
of equations representing a certain family of curves;
and any member of the family may be specified by the
particular value assigned to the constant c. The quan-
tity c, which is constant for the same curve but different
for different curves, is called the parameter of the
family.
138. Envelope. Definition.
Let all the members of the family of curves
ºb (a), y, c) = 0
be drawn which correspond to a system of infinitesimally
close values of the parameter, supposed arranged in
order of magnitude. We shall designate as consecutive
curves any two curves which correspond to two consecu-
tive values of c from the list. Then the locus of the
ultimate points of intersection of consecutive members
of this family of curves is called the ENVELOPE of the
family.
JEW VELOPES. I 63
139. The Envelope touches each of the Inter-
secting Members of the Family.
Let A, B, C represent three consecutive intersecting

A \
members of the family. Let P be the point of inter-
section of A and B, and Q that of B and 0.
Now, by definition, P and Q are points on the en-
velope. Thus the curve B and the envelope have two
contiguous points common, and therefore have ulti-
mately a common tangent, and therefore touch each
other. Similarly, the envelope may be shown to touch
any other curve of the system.
140. The Envelope of AX* + 2BX + C = 0 is
B” = A.C.
If A, B, C be any functions of w, y and the equation
of any curve be
AX* + BX + C = 0,
X being an arbitrary parameter, the envelope of all such
curves for different values of X is B" = A.C.
For the equation
AX +2BX + C = 0
may be regarded as a quadratic equation to find the
values of X for the two particular members of the family
which pass through a given point (a, y). Now, if (a, y)
be supposed to be a point on the envelope, these mem-
bers will in the limit be coincident. Hence for such
values of a, y the quadratic for X must have two equal
roots, and the locus of such points is therefore
B* = A.C.
11—2
164 DIFFERENTIAL CALCULUS.
Ex. 1. Thus the line y = ma; +. may be written in the form
m” – my + a = 0,
whence the equation of the envelope for different values of m is plainly
!/*= 4a2.
Ex. 2. The line y=mw-i-Naºmº-Flº
may be written in the form
m” (a.”— a”)–2may +y” — b%–0,
and the envelope is
aºy”– (v*—a”)(y?– b°),
Ol' a?/a2+ y”/b2–1.
141. Envelope of p (a, y, c) = 0.
The envelope of the more general family of curves
ºb (a, $/, c) = 0
may be considered in the same way.
It is proved in Theory of Equations that if f(c) = 0
be a rational algebraic equation for c the condition for a
pair of equal roots is obtained by eliminating c between
f(c) = 0 and f'(c) = 0.
Hence to find the envelope of
ºb (al, ſ, c) = 0
we differentiate with regard to c thus forming
0
Oc ºp (a, y, c) = 0
and then eliminate c.
Ex. 1. Thus to find the envelope of the line
Q,
y=cº-Hi
for different values of c, we have upon differentiating with regard to c
Q,
0=z-ji,
ENVELOPES. 165
whence cy = a + c2a: = 2a,
tº 9 (l, 4 2
and Squaring, y”. –a4,
giving y”= 4aa.
Ex. 2. Find the envelope of
a cos”0+y sin”6 = a
for different values of 6.
Differentiating with regard to 0,
— a cos”6 sin 0+y sin”0 cos 9=0,
* * cos 6 sin 9 Mcos39-Fsin?0 1.
glWing – = > ------------- E —- e
!/ (U Jºº-Hy? Jºº-Hy?
Hence the equation of the envelope is
3 3
Q? –*H + 1/ 4. 3. = 0,
(a^+ y”); (v*4-y”);
Ol' . –F#––a,
Jºº-Hy?
which may be written ! + * F J. ſº
a: y- a-
EXAMPLES.
1. Show that the envelope of the line
& 3/_
d + 5- l,
when ab=cº, a constant, is
4ay=c”.
2. Find the equation of the curve whose tangent is of the
form
3/= ma;+ m”,
m being independent of a; and y.
3. Find the envelope of the curves
(U' 3/
for different values of 6.
a2 COS 6 b% sin 6 gº
Cº.,
166 D/FFERENTIAL CA LOUI, U.S.
4. Find the envelope of the family of trajectories
3/=a tan 6–4, w”.
2” M2 COS2.6 °
6 being the arbitrary parameter.
5. Find the envelopes of straight lines drawn at right angles
to tangents to a given parabola and passing through the points
in which those tangents cut -
(1) the axis of the parabola,
(2) a fixed line parallel to the directrix.
6. Find the envelope of straight lines drawn at right angles
to normals to a given parabola and passing through the points
in which those normals cut the axis of the parabola.
7. A series of circles have their centres on a given straight
line, and their radii are proportional to the distances of their
corresponding centres from a given point in that line. Find the
envelope.
8. P is a point which moves along a given straight line.
PM, PN are perpendiculars on the co-ordinate axes supposed
rectangular. Find the envelope of the line M.V.
142. The envelope of
- $ (q, y, c) = 0..................... (1)
may also be obtained as follows:
Let ºb (a, V, c + 6c) = 0 .................. (2)
be the consecutive member of the family.
This, by Taylor's Theorem, may be expressed in its
expanded form as
0
ºb (w, y, o)+8 = b(º, y, c) + ... = 0.
Hence in the limit when Öc is infinitesimally small
we obtain
0
ă, b(º, $/, c) = 0
as the equation of a curve passing through the ultimate
point of intersection of the curves (1) and (2),
EWVELOPES, 167
If then we eliminate c between
q (a, y, c) = 0
0 y
and 0c © (w, y, c) = 0
we obtain the locus of that point of intersection for all
values of the parameter c. This is the Envelope.
Polar curves of the form
$ (r, 0, c) = 0
may be treated in the same manner.
Ex. 1. Find the envelope of a circle drawn upon radii vectores of
the circle r=2a cos 0 as diameters.
Let d, a be the polar co-ordinates of any point on the given circle,
then d=2a cos a.
The equation of a circle on the radius vector d for diameter is
r=d cos (6 - a).............................. (1),
3]." r=2a cos a cos (0- a) ..................... (2).
Here a is the parameter. Differentiating with regard to a,
— sin a cos (9– a) +cos a sin (0- a)=0,
whence sin (9–2a)=0 or a = . s & © tº * * * * * * * * * * * * * * * * * (3).
Substituting this value of a in (2),
q = 24 cos; or r=a. (1+cos 0),
the equation of a cardioide.
Ex. 2. Find the envelope of a straight line drawn at right angles
to radii vectores of the cardioide r = a (1 + cos ?) through their ex-
tremities.
Here we are to find the envelope of the line
a cos a + y sin a = d,
where d, a are the polar co-ordinates of any point on the cardioide;
i. e. where d = a (1 + cos a).
The equation of the line is therefore
a cos a +y sin a = a (1 + cos a),
168 DIFFERENTIAL CA LOUI, U.S.
Oſ (a – a) COs a + y sin a = a,
a line which from its form obviously touches the circle
(a - a)*+y}=a” or r=2a cos 0,
which is therefore its envelope.
Ex. 3. Find the envelope of
a || _
Fji= 1 ................................. (1),
with the condition
a”--b"=constant =c" say ..................... (2).
Here there are two parameters with a condition connecting them, so
that only one is independent. Imagine a and b to be both functions
of some third arbitrary parameter t. Differentiating both equations
with regard to t,
a da y do
a2 di Tiº di Tº"
db
da
ºv–1 lº *~l ... — ():
(l, dt + b dt 0;
4 y * Lºſ
a 0 a b l
ve, 7–7 =:::H7 ==
Thus aºrti = cºa, and bºt' = c^y.
... substituting in (2),
—“– _*
(c” a)*H + (cºy)*H = c”,
*_ _ " - ??
Or aft-H + yºti - cºtl,
which is the required envelope.
EXAMPLES.
1. Find the envelope of
3) =ma – 2am — amº,
i.e. of normals of the parabola
3/*=4aa.
EXAMPLE'S. 169
2. Find the envelope of
C!!! by $3 P &
- a” — b%,
*
cos 67 sin 67
*.e. of normals of the ellipse
aft/a2+y}/b}=1.
3. Find the envelopes of
- (a) y=mw-Ham”,
(3) y=ma;+ amº,
(y) y= ma + am”;
7m being the arbitrary parameter.
4. Find the envelopes of
(a) a cos 6-1-ysin 6=a,
(8) w Woos 0+y Wsin 9=a,
3} 2
9 via twin-"
(8) w cosá 64-ysiná 6–a,
(e) aſ cosł64. A siné6–a,
(Ö) a cost 6+y sin” 6–a,
(m) a cos”6+y sin”6=a;
6 being the arbitrary parameter in each case.
5. Find the envelopes of
(a) w cos a+ysin a-a Woos 2a,
(8) a COS2a+/sin 20 = a cos”a,
º
(y) a cosma-Hysin ma=a. (cos na)";
a being the parameter in each case.
6. Find the envelopes of circles described on the radii vectores
of the following curves as diameters:— *
(a) aº/a2+y^ſb%–1,
(3) y”=4aw,
I70 DIFFERENTIAL CALCULUS,
(y) y”=4a (a + a),
(8) r=a. (1+cos 6),
(e) :=14, cos 6,
(...) rºcos 20=a”,
(m) rº–a” cos 20,
(6) r" – aſ cosmó.
7. Find the envelopes of the circles which pass through
the pole and whose centres lie on the several curves of ques-
tion 6.
8. Find the envelopes of straight lines at right angles to
the radii vectores of the following curves drawn through their
extremities:–
(a) a straight line,
(8) a circle through the pole,
(y) any circle,
- (8) a cardioide r- a (1+cos 6),
(e) a limaçon r=a+b COs 6,
(6) a lemniscate rº–a’ cos 26,
(m) an equiangular spiral r=ae”.
(6) * = 0." COS mé.
9. Find the envelopes of the straight line
* L3/_
g-F#– l
under the several conditions
(a) a + b =c,
(8) a” +b^ = c”,
(y) amb” = 6m + 7.
EXAMPLE'S. - I7]
10. Find the envelopes of the ellipse
a;2 y?
… + , =l
under the several conditions
(a) a”--b%= cº,
(6) Cºl. + bn -: cº,
(y) gºrºbº = Gºvt *.
11. Find the envelopes of the parabola
* //
V; V-
under the several conditions
(a) a + b =c,
( 8) a” + bº = cº,
(y) cºmbº = 6m + v.
12. Find the envelope of
&m 7)?,
*4-4-1
(ºr, bºn
under the condition
ap+bp = cº.
13. Show that the envelope of the family of curves
AX*-ī-3BA%+30) + D=0,
where A is the arbitrary parameter, and A, B, C, D are functions
of a; and y, is
(BC–AD)}=4 (BD – C*) (AC-B”).
14. If A, B, C be any functions of a; and y the envelope of
º,
A cos ma-H B sin ma= 0(cos na)”
(* + *
tan - 1 B *
72,
cos” 72,
7?? — A.
Toº
15. A straight line of given length slides with its extremities
on two fixed straight lines at right angles. Find the envelope
of a circle drawn on the sliding line as diameter,
172 DIFFERENTIAL CALCULUS,
16. Show that the envelope of straight lines joining the
extremities of a pair of conjugate diameters of an ellipse is a
similar ellipse. -
17. The envelope of the polars with respect to the circle
*=2a cos 6
of points which lie upon the circle r-2b cos 6 is
{(a –b) wi-ab}=b” {(a - a)2+y}}.
18. Show that the pedal equation of the envelope of the
line -
a cos má--ysin m3 = a cosmó
is mºr” –(m”—n?)p?--n}a}.
19. Two particles move along parallel lines, the one with
uniform velocity and the other with the same initial velocity
but with uniform acceleration. Show that the line joining them
always touches a fixed hyperbola.
20. Show that the radius of curvature of the envelope of
the line
a cosa-i-ysin a = f(a)
is f(a)+f"(a),
and that the centre of curvature is at the point
a = – f'(a) sin a – f'(a)cos |
3/= f'(a) cosa-f"(a) sin aſ
CHAPTER, XII.
ASSOCIATED LOCI.
143. IT is intended in the present chapter to present
a brief introduction to a study of several important loci
which are intrinsically connected with every curve.
PEDAL CURVES.
144. DEF. If a perpendicular 0Y be drawn from
any fixed point 0 upon the tangent to any curve the
locus of Y is called the “first Positive Pedal” of the
original curve with regard to the given point 0.
Two important cases occur in the Conic Sections:
(1) In a parabola the locus of the foot of the per-
pendicular from the focus upon a tangent is
the tangent at the vertex. This line is there-
fore the first positive pedal of the parabola
with regard to the focus.
(2) In a central conic the locus of the foot of the
perpendicular from a focus upon a tangent is
the auxiliary circle. This circle is therefore
the first positive pedal of the conic with regard
to the focus. *
145. To find the Pedal with regard to the
Origin for Cartesian Curves.
When the Cartesian equation of a curve is given,
the condition of tangency of
Y cos & + Y sin a = p
174 DIFFERENTIAJ, CA LOUI, U.S.
may be obtained by comparison of this line with the
tangent at any point (a!, y).
Let this condition when found be
f(p, a) = 0.
Then since p, a are the polar co-ordinates of the point
whose locus is sought we may replace them by the
current co-ordinates r, 6, and the equation of the pedal
will be
f(r, 0) = 0.
Ex. 1. The condition that
a cos a + y sin a = p touches aſ”|a”--y”|b°= 1
is known to be p?= a” cos” a + b% sin” a.
Hence the first positive pedal with regard to the origin is
T*= a” cos26 -- b” Sin?6.
Ex. 2. Find the first positive pedal of the curve aſ "y” – a "t" with
regard to the origin.
The equation of the tangent is plainly
, Til , \, n.
Y--- Y - = m + m.
3/
£
Comparing with X cos a + Y sin a = p,
a cos a y sin a p
in n m + n’
e - T]l, p 7l, p
glving (C E º 3. y = −. ---.
777 -- 77 COS d. 711 - ?? S11] O.
Hence the condition of tangency is
- 7)?. º,
& tº P * P =amºtº,
7), +.77 COS d. 777 - ?), S11, d.
and replacing p and a by r and 6, the equation of the pedal becomes
(m + m)”, ©
..— ... ſºns??? 27,
mºnum? COS''' {} Sill 0.
jºin-Hw – (lºw-Fly
146. To find the Pedal with regard to the
Pole of any curve whose Polar Equation is
given.
PEDALS, 175
Let - F'(r, 0) = 0 ..................... (1)
be the equation of the curve.

|-
Let r", 6" be the polar co-ordinates of the point Y,
which is the foot of the perpendicular 0Y drawn from
the pole on a tangent. Let OA be the initial line.
Then
A A A
6 = A OP = A OY -- YOP
= 0 + - d … (2),
also tan q = r º tº tº e º is s g g s is g g s w tº º $ 9 s tº e g g s & 8 tº $ tº $ 4 & e (3),
and r' = r sin (p,
1 – 1 - 1 (%) (Art. 95) ........(4).
OT 7'2 T 12 7.4 Volò
If r, 6, # be eliminated from equations 1, 2, 3, and 4,
there will remain an equation in ", 6". The dashes
may then be dropped and the required equation will be
obtained. -
Ex. To find the equation of the first positive pedal of the curve
** = aſhº COS ºn 6.
Taking the logarithmic differential
7m dº'
7 diffT m tan muff;
therefore cot q = -- tan m{};
therefore ºp= ; + m3.
176
DIFFERENTIAL CALCULUS.
But
therefore
Again
T
{}= {}' + 2. cº-sºs @,
f
0–0' – mb, or 6 = −.
m + 1
r' = rsin ºp-r cosmó
l
= a COS" m0. COS m{}
Hence the equation of the pedal curve is
147. DEF. If there be a series of curves which we
_Tº_ Tº 7)?
ºn-F1 = a'm-Fl COS —- ().
ºn + 1
may designate as
such that each is the first positive pedal curve of the
one which immediately precedes it ; then Aa, Aa, etc.,
are respectively called the second, third, etc., positive
pedals of A. Also, any one of this series of curves may
be regarded as the original curve, e.g. As ; then A, is
called the first negative pedal of As, A
tive pedal, and so on.
4, 41, 42, 43, 4n, ...
Ex. 1. Find the k” positive pedal of
7" - aſtrº COS m{}.
It has been shown that the first positive pedal is
where
r” =a^1 cosm, 6,
777,
77b1 = −- .
*T1-Hºm
Similarly the second positive pedal is
r”2=a” cosm, 9,
711, 711 m .
*T1-Em, T 1-E2m."
where
and generally the k" positive pedal is
where
** = a” cosm;6,
777,
}}l k = 1-1 - .
1 + km.
1 the Second mega-
PEDALS. 177
Ex. 2. Find the kº negative pedal of the curve
* = 0.77% COS mé).
We have shown above that r"-a" cosmó is the kºh positive pedal
of the curve r"= a” cosmē, provided m==*-.
1 + km.
1–. "
T 1 – km
Hence the kº negative pedal of r"= a” cos m0 is
This gives 7
*= a” cosmö,
7??,
where 70 - 2–3–-- .
1 — km.
EXAMPLES.
1. Show that the first positive pedal of a circle with regard
to any point is a Limaçon (r-a+b cos 6), which becomes a
Cardioide {r=a. (1+cos 6)} when the point is on the circum-
ference.
2. Show that the first positive pedal of a central conic with
regard to the centre is of the form tº- A + B cos 26, which
becomes a Bernoulli’s Lemniscate (r” – a” cos 26) when the conic
is a rectangular hyperbola.
3. Show that the first positive pedal of the parabola y”=4aa,
with regard to the vertex is the cissoid
w (a:4+?)--ay?=0.
4. Show that
# # 2 # # 2
5
*= a” cos 26, r* = a cos 36. 7" - (, cos 56.
«» Q o
# # 2 , # # 2
r' =a' cos #6, r* = a” cos .. 6
7 9 * *
are the first, second, third, fourth and fifth pedals of a rect-
angular hyperbola.
5. Show that the 10th positive and negative pedals of the
circle r- a cos 6 are respectively t
6 6)
tº --— s11 * – ,9
7 – a cos" - , and 7'- a secº = .
II 2 9
178 D/FFEREWTIAL CALO/UI, U.S.
6. Show that the first positive pedal of
Aft +- ſy” Fº a",
7? 78. 72
78.
is (2+y})*1 = air-i (wº-it-yº-i).
148. Perpendicular on Tangent to Pedal.
Let PY, QY’ be tangents at the contiguous points
P, Q on the curve, and let OV, 0Y" be perpendiculars
from 0 upon these tangents. Let 0Z be drawn at right
angles to YY produced. Let the tangents at P and Q
intersect at T.

It is clear that since
yöy- yīy”,
the points O, Y, Y', T are concyclic, and therefore
OYZ-T- of y' = Offy";
and the triangles 0YZ and 0TY" are similar. Therefore
OZ OY".
OW TOT.
and in the limit when Q comes into coincidence with P.
Y' comes into coincidence with Y, and the limiting
PEDAJAS. 179
position of YY is the tangent to the pedal curve. Let
the perpendicular on the tangent at Y to the pedal
curve be called p, then the above result becomes
Pi P
p : ’
OI’ pºr = p".
149. Circle on Radius Vector for diameter
touches Pedal.
This fact is clear from the figure of Art. 148, for OT
is in the limit a radius vector and the circle on OT' for
diameter cuts the pedal in the ultimately coincident
points Y and Y', and therefore in the limit has the
same tangent at Y as the pedal curve.
150. Pedals regarded as Envelopes.
It is clear then that the problem of finding the first
positive pedal of a given curve is identical with that of
finding the envelope of circles described on radii vec-
tores as diameters (see Art. 142).
Again, the first negative pedal is the curve touched
by (i.e. the envelope of) a straight line drawn through
any point of the curve and at right angles to the radius
vector to the point.
Thus by Art. 142, Ex. 1, the first positive pedal of
* = 2a. COs 6
with regard to the origin is the cardioide
r= a (1 + cos 6),
and by Ex. 2, the first negative pedal of
r= a (1 + cos ?)
with regard to the origin is the circle
r = 2a coS 6.
180 DIFFERENTIAL CA LOUI, U.S.
INVERSION.
151. DEF. Let 0 be the pole, and suppose any
point P be given; then if a second point Q be taken on
OP, or OP produced, such that O.P. OQ = constant, k”
say, then Q is said to be the inverse of the point P with
respect to a circle of radius k and centre 0 (or shortly,
with respect to 0).
If the point P move in any given manner, the path
of Q is said to be inverse to the path of P. If (r. 6) be
the polar co-ordinates of the point P, and (r', 6) those
of the inverse point Q, then
q',' = k”.
Hence if the locus of P be
f(r, 6) - 0,
f(;, 0)-0
For example the curves
that of Q will be
ºn' – a nº cos m.9 and r* COS mt) = a”
are inverse to each other with regard to a circle of radius a.
152. Tangents to Curve and Inverse inclined
to Radius Vector at Supplementary Angles.
If P, P be two contiguous points on a curve, and

INVERSION. 181
Q, Q the inverse points, then, since
OP. OQ = OP’. OQ',
the points P, P, Q, Q are concyclic; and since the
angles OPT and 00'T' are therefore supplementary, it
follows that in the limit when P' ultimately coincides
with P and Q with Q the tangents at P and Q make
supplementary angles with OPQ.
EXAMPLES.
1. Show that the inverse of the conic
:- 1 + e COS 6
with regard to the focus is the Limaçon
- #r-1+ecoso
which becomes a cardioide
[r=a. (1+cos 6)]
in the case when the conic is a parabola.
2. If the point a', y' be inverse to (w, y), show that
Twº-Eyº
3. Show that the inverses of the lines w = a, y = b are re-
spectively
k2
&;2 +y” - a 30,
Å;2
and - a 24-y?= 7. /.
4. Show that the inverse of the conic
aw”--2bay--cy”=2/,
is the cubic /*(aw”--2bºy--oy")=2/ (w84-y”).
5. Find the inverses of the straight lines
3a;+4y=5, 4a – 3/=5
with regard to the origin, and show that they are circles cutting
Orthogonally.
182 DIFFERENTIAL CALCULUS,
POLAR RECIPROCALS.
153. Polar Reciprocal of a Curve with regard
to a given Circle.
DEF. If 0Y be the perpendicular from the pole
upon the tangent to a given curve, and if a point Z be
taken on OY or 0Y produced such that 0Y. OZ is con-
stant (=k* Say), the locus of Z is called the polar reci-
procal of the given curve with regard to a circle of
radius k and centre at 0. -
From the definition it is obvious that this curve is the
inverse of the first positive pedal curve, and therefore its
equation can at once be found.
Ex. Polar reciprocal of an ellipse with regard to its centre.
a;2 g”
For the ellipse a? + is = 1,
the condition that p = a, cosa + y sina touches the curve is
p°– a” cos” a + b% sin”a.
Hence the polar equation of the pedal with regard to the origin is
79– a 2 GOS26 -- b2 Sin?6.
Again, the inverse of this curve is
l;4 g
... = a” cos”0+ b” sin” 9,
Ol' a*a*4-b”y?=k*,
which is therefore the equation of the polar reciprocal of the ellipse
with regard to a circle with centre at the origin and radius k.
EXAMPLES.
Find the polar reciprocals with regard to a circle of radius k
and centre at the origin of the curves. *
1. 7'- a cos 6.
Any circle.
** = a” COS 7.6.
amy” = 0 m + 7t.
2.
q;” + y” - Cº.
4. 7)?, 3/ m_
(...) + () = 1.
7"
l
3. – = 1 +e cos 6.
4.
r= a (1+cos 6).
IV VOLUTES AND EVOLUTES. 183.
INVOLUTES AND EVOLUTES.
154. DEF. The locus of the centres of curvature
of all points on a given curve is called the evolute.
If the evolute itself be regarded as the original
curve, a curve of which it is the evolute is called an
involute.
Ex. To find the evolute of the parabola
gº–4aa.
By Ex. 2, p. 153, the co-ordinates of the centre of curvature are
㺠= 2a +3a;
7= "...] o
The arbitrary abscissa a must be eliminated between these equations.
We have
(3–2a)}=3344–-3%a}|2,
or squaring and dropping the bar,
- 4 (a – 2a)*=27ay”,
a semicubical parabola.
EXAMPLES.
1. Show that the locus of the centres of curvature of the
ellipse
wº/a2 +y}/b}=1,
is (aw)”--(by)*=(a”—b%)”.
2. Show that the locus of the centres of curvature of the
catenary
(U
3/= c cosh a
of2 — Aa2
c log º46%
i. = n+% w/73–43
1S 2 wº. Wy 46%.
155. Evolute touched by the Normals.
Let Pi, P, P, be contiguous points on a given curve,
and let the normals at Pi, P, and at P, P, intersect at
Q, Q, respectively. Then in the limit when P, P,
184 JDIFFERENTIAJ, CA LOUI, U.S.
move along the curve to ultimate coincidence with P.
the limiting positions of Q, Q, are the centres of curva-

P;
P.
ture corresponding to the points Pi, P, of the curve.
Now Q, and Q, both lie on the normal at P., and there-
fore it is clear that the normal is a tangent to the locus
of such points as Q, Q, i.e. each of the normals of the
original curve is a tangent to the evolute; and in general
the best method of investigating the equation of the
evolute of any proposed curve is to consider it as the
envelope of the normals of that curve.
Ex. 1. To find the evolute of the ellipse
a2 y?
is +;=1.
The equation of the normal at the point whose eccentric angle is
q is
(lſº by
cos q sin ºp
We have to find the envelope of this line for different values of the
parameter p.
Differentiating with regard to p,
sin ºp COS ºp *
!! --R-T = U. . . . . . . . . . . . . . . . . . . . . . . 2),
*cºat l sin” (p () (2)
sin"d cos’ſ
by " Tar T
Ol' 0.
sin ºf eos @ 1
- 3/7– - 3 y— = -F—-
- Why War VºItºi
Substituting these values of sin p and cos p in equation (1) we obtain,
! sº a tº 2. 2 2.
after reduction (aa)3 + (by)3 = (a” – U2)3.
Hence
tº s e º e º - 4 - e a s e s - - - - (3).
IV VOLUTES AND EVOLUTES. 185
Ex. 2. Shew that the envelope of y = ma - 2am — am” (i.e. the
normal to y”= 4aa) is 27ay”=4 (a – 2a)”.
156. There is but one Evolute, but an infinite
number of Involutes. -
Let ABCD... be the original curve on which the
successive points A, B, C, D, ... are indefinitely close to
each other. Let a, b, c, ... be the successive points of
intersection of normals at A, B, C, ... and therefore the
centres of curvature of those points. Then looking at
ABC... as the original curve, abod... is its evolute.
And regarding abcd... as the original curve, ABCD...
is an involute.
O
If we suppose any equal lengths A A', BB, CC", ...
to be taken along each normal, as shown in the figure,
then a new curve is formed, viz. A'B'C'..., which may
be called a parallel to the original curve, having the
same normals as the original curve and therefore having
the same evolute. It is therefore clear that if any

186 DIFFERENTIAJ, OA LC UI, U.S.
curve be given it can have but one evolute, but an
infinite number of curves may have the same evolute,
and therefore any curve may have an infinite number of
'nvolutes. The involutes of a given curve thus form a
system of parallel curves. -
157. Involutes traced out by the several points
of a string unwound from a curve. Length of
Arc of Evolute.
Since a is the centre of the circle of curvature for
the point A (Fig., Art. 156),
a.A = a B
= b b + elementary arc ab (Art. 81).
Hence a.A. – bl: = arc ab. -
Similarly bF – c.6 = arc be,
cC – d.1) = arc ca,
etc.,
ff"—gG = arc fg.
Hence by addition
a.A – g(# = arc ab + arc be + ... + arc fg
= arc ag.
Hence the difference between the radii of curvature at
two points of a curve is equal to the length of the corre-
sponding arc of the evolute. Also, if the evolute abc...
be regarded as a rigid curve and a string be unwound
from it, being kept tight, then the points of the unwind-
ºng string describe a system of parallel curves, each
of which is an involute of the curve abcd..., one of
them coinciding with the original curve ABC.... It is
from this property that the names involute and evolute
are derived.
EXAMPLE'S. 187
EXAMPLES.
1. Show that the whole length of the evolute of the ellipse
wº/a2+y}/b}=1,
e 0.2 b2
1S 4(;-).
2. Show that in the parabola
3/*=4aw
the length of the part of the evolute intercepted within the
parabola is
4a (3 W3–1).
EXAMPLES.
1. Show that the fourth negative pedal of the cardioide
*= a (1+cos 6)
is a parabola.
2. Show that the fourth and fifth positive pedals of the
CUIrVe
9
2 \?
7" = 0, (*#)
3.I’ê respectively a rectangular hyperbola and a lemniscate
[r?–d?cos 29].
3. Show that the first positive pedal may be obtained by
.2
writing rinstead of p and º instead of r in the pedal equation of
the original curve.
4. Show that the first positive pedal of the epicycloid
p?= Ar” +B,
has for its pedal equation
7.4
7% = A 72 + B.
5. Show that the equation of the nth pedal of the curve
f (p, r)=0,
e 7”. ſºn + 1
|S f(ºr +)-0.
188 JDIFFERENTIAL CALOUI, U.S.
i.e. it may be obtained from the original equation by writing
* º, 7" º,
- for p, and () 7" for 7.
(..) p Ior p p
6. Show that the inverse of the hyperbola
- 1 1 1
* - *
w" y Ta
with regard to the origin is
a (a^+y}) (a +y)=k*ay.
7. Show that the inverse of the ellipse
a;2 y?
... + =l,
with regard to the origin is
2 a.2
(º-Fy?)?=} (.4 $/ ) tg
2 T 52
8. Show that the equation of the inverse of a curve with
regard to the pole may be obtained from the pedal equation
by writing
}% £2
T for p and 7 for r,
i.e. if f(p, r)=0
be the original curve
Å;2 Å;2
f (* 5 £) = 0
7
will be the inverse.
9. Show that the inverse of the curve
p°– Ar” + B,
is Kºp”—Akrº-H Brº.
10. Show that the pedal equation of the evolute of the
curve p-f(r) will be obtained by eliminating p and r between
- ſo =f (r),
p?=1%–p”,
dr
... 2–2 ...? ("\"
ri” =7%+r dp -**
Jº YAMPLE'S. 189
11. Show that the evolute of the epi- or hypocycloid
p?– Ar” + B
is p?a=Ar” + #(4–1)
(i.e. a similar epi- or hypocycloid, see Q. 23, p. 113).
12. Prove the following series of results for the Equiangular
or Logarithmic spiral
6 Coto.
7" – O.6 e
(1) ºp-a. (Hence the name “Equiangular.”)
(2) The pedal equation is
p=7 sin a.
(3) p = 7' coSec a.
(4) Let 0 be the pole, PT the tangent at P, OY the
perpendicular, OT the polar subtangent cutting the normal in C.
Prove that C is the centre of curvature.
(5) 8–7 sec a (3 being supposed measured from the pole).
(6) s—PT.
(7) Prove geometrically that all the pedals, the inverse,
the polar reciprocal, the evolute are equal equiangular spirals
(i.e. for which q is the same).
(8) The equation of the first positive pedal is
Tr
*= a sin a (;-) “, cota
(9) The nth pedal is
Tr
r=a sin" a e" (;-) * 2° cot a.
(10) The inverse is
Å;2 – 6 cot a
** = – 6 g
(11) The Polar reciprocal is
Tr
Å;2 -(;- ) cote 2-0cot a
7" - —--—--—
a sin a
190 DIFFERENTIAI, OAJCULUS,
(12) The evolute is
T COt
* * OL
r=a cotae 29 cot a .
13. Prove the following series of results for the cardioide
* = a (1 – cos 6). - -
(1) The curve may be constructed as the locus of a point
on the circumference of a circle of diameter a which rolls without
sliding upon the circumference of a circle of equal radius.
(2) Hence prove geometrically
6
4–3.
Prove this also from the equation by means of the formula
dé
tan q =r ºr "
(3) If 0 be a fixed point upon a circle of radius a and
centre C, and P any other point upon the circumference, make
A
the angle cho-cho. Prove that PQ always touches a car-
dioide formed by the rolling of a circle of radius a upon a circle
3
of equal radius (geometrical).
(4) The curve may also be constructed thus:–Take a
circle OQD of diameter a and centre E. Let a straight rod PP"
of length 2a move in such a manner that its mid-point Q de-
scribes the given circle whilst the rod is constrained to pass
through a fixed point 0 on the circumference. The points P, P’
trace out the cardioide. The point O may be called the focus.
(5) Any “focal chord” is of constant length.
(6) The “Instantaneous Centre” for the motion of the
rod is at the point R of the circle in (4) diametrically opposite
to Q. - -
(7) The lines RP, RP are normals (geometrical).
(8) Normals at the ends of a focal chord intersect at right
angles on the circle in (4) (geometrical). -
(9) Tangents at the ends of a focal chord intersect at
right angles on a concentric circle of three times the radius
(geometrical). t
(10) If R P cuts the circle in (4) again at S the angle OSR
is bisected by ES (geometrical).
FXAMPLES. 191
(11) Hence show (by 3) that the evolute of the cardioide
is a cardioide of one-third the linear dimensions and turned the
opposite way (geometrical).
(12) Show (by 11) that the whole length of the arc of
the cardioide
*= a (1 — cos 6) is 8a.
(13) The cardioide is the first positive pedal of a circle
with regard to a point upon the circumference.
(14) The pedal equation is
;3
ſo (2a)*
(15) The curvature at any point is
* 3/2 W2ar.
(16) The nth pedal of
*= a (1+cos 6),
1 }
1S r” +2= (2a)” tº cos n+2
(17) The Inverse of the cardioide with regard to the pole
is a parabola.
14. Show that if p be the radius of curvature at any point
p, r upon the curve f(p, r)=0 and pi that at the corresponding
point upon the inverse, then
-*/(ºr -º),
PI ºp p
where k” is the constant of inversion.
15. With the same notation the radius of curvature at the
corresponding point of the polar reciprocal is
Å;27.3
3
pºp
where k” is the constant of reciprocation.
5
16. With the same notation if pn be the radius of curvature
at the corresponding point of the nth pedal, prove that
gºv-2 | 1 || 1 pp
Pn ºn-i ºf 3 o 5 * *-*.
p 2–2–2 – ... – 2 – 7%
(where there are n + 1 quotients)
nº-(n-1) pp
T (n+1)79–7pp
CHAPTER XIII.
MAXIMA AND MINIMA. ONE INDEPENDENT WARIABLE.
158. Elementary Algebraical and Geome-
trical methods.
Examples frequently occur in Algebra and Geometry
in which it is required to find whether any limitations
exist to the admissible values of certain proposed
functions for real values of the variable upon which
they depend. These investigations can often be con-
ducted in an elementary manner. A few examples
follow in illustration.
Ex. 1. The function aa; + º may be written in the form
(Jay- V) +2 Vab,
from which it is obvious that the expression can never be less than
2 Jab, the value it attains when Vaw- V º Or a = }. For
the square of a real quantity is essentially positive and therefore any
value of a; other than a = V } will give to the expression a greater
value than 2 W ab.
Ex. 2. Investigate whether any limitation exists to the real values
3a.” – 4a: +3
of the expression 32-H42-E3 for real values of a.
3ac” – 4a: +3
Let 㺠IIA, IBT 9.
Then 3 (1-y) wº–4 (1+y) a + 3 (1 – y)=0.
MAXIMA AND MINIMA. 193
If a be real, we must have - -
4 (1+y)*–9(1-y)*> 0,
i.e. (5) – 1) (5 – y) must be positive.
Hence y must lie between the values 5 and . e
Therefore the maximum value of the expression is 5 and the
* & . 1
minimum value is 6
Ex. 3. If the sum of two quantities be given, when is their pro-
duct a maximum? -
Let a + y = a, a constant,
then 4ay= (a + y)*— (a) – y)*= a” - (a — y)*.
The right-hand side has its maximum value when (a, -y)” has its
minimum, i.e. when a = y, for being a square it cannot be negative.
2
Thus the maximum value of ay is * §
This may be shown geometrically as follows:–the problem is to
divide a given line AB in such manner that the rectangle of the
segments is as great as possible. Let C be the centre and P any other
point of the line. Then by Euc. II. 5,
rect. AP. PB + sq. on CP=Sq. on AC =rect. AC. CB,
i. e. the rect. AP. PB is less than the rect. AC. C.B.
Hence the point of division is the mid-point, or the line must be
bisected.
Ex. 4. If a + y + 2 + w be constant, when will ayzw have its
maximum value?
So long as any two, say a and y, are unequal we can without
altering 2 and w (and thus keeping a +y constant) increase ay, and
therefore also a y2w by making a, and y more nearly equal (by Ex. 3).
Hence ayzw does not attain its maximum value until
a = y = 2 = w.
The same argument obviously applies to the product of any number
of quantities whose sum is constant.
If we are searching for the maximum value of such an expression
as a y”z”, say, with condition a + y + 2 = a, we proceed thus
a y”z*=2°3°. a. . . . . . . . . . . and is to be a maximum,
- ! 19 . . . * * *—, .
where * + 3 + 3 + 3 + 3 + 3 +4.3
E. D, C, 13
194 DIFFERENTIAJ, CA LOUI, U.S.
and by the preceding work we are to make
3. {} & a
IT 3 T 3 T 6
22. 33a 6 aff
whence the maximum value is TööT = 21.35
Ex. 5. In any triangle the maximum value of
cos A cos B cos C is .
For 2 cos A cos B cos C=cos A {cos (B-C) – cos A},
and therefore as long as B and C are unequal we may increase the
expression by making them more nearly equal and keeping their sum
constant. Thus cos A cos B cos C does not attain its maximum value
until A=B=C =; , and then its value is . e
Ex. 6. What are the greatest and least values of a sin a + b cosa!?
Let a = c cos a and b = c sin a,
so that c”= a” + b% and tan a=}.
Thus a sin a + b Cosa:= c sin a cos a + c cosa; sin a
= VažTº sin (a +a),
and as the greatest and least values of a sine are 1 and – 1, the
maximum and minimum values required are Vaº-FU3 and – Vaºbº
respectively.
Ex. 7. If A, B, C... be a number of points and P any other
point, and if G be the centroid of masses X at A, u at B, etc., then it
is a known proposition that -
XPA3+APB^+... =(AGA*4 p.GB-4...)+(A+,+...) PG",
Or - XPA*=XXGA2+ (XX) PG?.
Hence since 2XGA* is a fixed quantity for all positions of P, XXPA2
has its minimum value when P is at G.
EXAMPLES.
1. Show that the minimum value of
ac”— 44; +9 is 5.
2. Show that the expression •+. cannot have any value
intermediate between 2 and -2,
EXAMPLE'S. 195
* +a;+ 1
3. Show that a 2–a–E I
has 3 for its maximum value, and
I g :--> , ,
3. for its minimum.
22
4. Show that the value of a”--pa;+1 is intermediate be-
a!”—pa;+ 1
tween
2+p 2–p
2–p and 2+p
a&%+ 2b^2 + c . tº - t -
-------- 2 if
5. Show that ca”--2ba;+ a is unlimited in value i
a + c <2b.
6. Show that the shortest distance from a given point to a
given straight line is the perpendicular distance.
7. Show that the greatest triangle inscribed in a given circle
is equilateral. -
8. Deduce from 7 by projection that a maximum triangle
inscribed in an ellipse
(a) is such that the tangent at each angular point is
parallel to the opposite side,
(b) has its centroid at the centre of the ellipse,
3 W3
4 ab, where a and b are the semi-
(c) that its area=
3, X6S.
9. Show that the triangle of greatest area with given base
and vertical angle is isosceles.
10. Show that if ABC be a triangle, and P any point
PA2+ PB2+ PC?
will be a minimum when P is at the centroid.
11. Show that
PA” tan A + PB% tan B-F PO% tan C.
has its minimum value when P is at the Orthocentre.
12, Show that
PA2 sin A + PB2 sin B+ P02 sin C
has its minimum value when P is at the incentre.
13–2
196 DIFFERENTIAL CALOUI, U.S.
13. If ABCD be a quadrilateral, and P any point,
PA2+ PB2--PO2+ PD2
has its minimum value when P is at the intersection of the
joins of mid-points of opposite sides.
14. Find the maximum rectangle inscribable in a given
ellipse, i.e. find the maximum value of ay, having given
a?/a2+y}/b}=1.
15. Find the maximum value of ayz, having given
a;+/-H2=a.
16. Find the maximum value of ayz, having given
17. Find the maximum value of ay”, having given
w--y=a.
18. Find the maximum value of a Pºyº, having given
a +3/=a.
19. If 6-1-q) = constant, the maximum value of sin 6 sin q is
attained when 6= q.
20. Find the maximum value of
sin A sin B sin C
for a triangle.
21. Find the maximum and minimum values of
a sin” a + b cos” a.
22. Show that the greatest chord through a point of inter-
section of two given circles is that which is parallel to the line of
centres. -
23. Find the greatest triangle of given species whose sides
pass through three fixed points.
24. Find the greatest rectangle whose sides pass through the
angular points of a given rectangle. -
25. Find the two perpendicular focal chords of a given conic
whose sum is a maximum.
MAXIMA AND MINIMA. 197
THE GENERAL PROBLEM.
159. Suppose a to be any independent variable
capable of assuming any real value whatever, and let
ºb (a) be any given function of w. Let the curve
y = b(a) be represented in the adjoining figure, and
let A, B, C, D, ... be those points on the curve at
which the tangent is parallel to one of the co-ordinate
3, XCS.
O| Mt M2 X
Suppose an ordinate to travel from left to right
along the axis of w. Then it will be seen that as the
ordinate passes such points as A, C, or E it ceases to
$ncrease and begins to decrease; whilst when it passes
through B, D, or F it ceases to decrease and begins to
tncrease. At each of the former sets of points the
ordinate is said to have a maximum value, whilst at
the latter it is said to have a minimum value.
160, Points of Inflexion.
On inspection of the accompanying figure it will be at
once obvious that at such points of inflexion as G or H,

198 DIFFEREAVTIA/, OAJLC UI, U.S.
where the tangent is parallel to one of the co-ordinate
axes, there is neither a maximum nor a minimum or-
21
O N X
dinate. Near G, for instance, the ordinate increases up
to a certain value NG, and then as it passes through
G it continues to increase without any prior sensible
decrease.
This point may however be considered as a com-
bination of two such points as A and B in the figure of
N1 N2
Art. 159, the ordinate increasing up to a certain value
N.G., then decreasing through an indefinitely small and
negligible interval to N.G., and then increasing again
as shown in the magnified figure, the points G, G, being
ultimately coincident.
161. We are thus led to the following definition:-
DEF. If, while the independent variable a, increases
continuously, a function dependent upon it, say ‘b (a),

MAXIMA A.VD MIVIMA. 199
ºncrease through any finite interval however small until
a = 0, and them decrease, b (a) is said to be a MAXIMUM
value of b (a). And if p (a) decrease to 4 (a) and them
increase, both decrease and increase being through a
finite interval, then b (a) is said to be a MINIMUM value
of b (a).
162. Criteria for the discrimination of Maxima.
and Minima, Values.
The criteria may be deduced at once from the aspect
d du . tº . . ) g
of º as a rate-measurer. For % is positive or negative
according as y is an increasing or a decreasing function.
Now, if y have a mainimum value it is ceasing to in-
crease and beginning to decrease, and therefore dy
da,
must be changing from positive to negative; and if y
have a minimum value it is ceasing to decrease and
beginning to increase, and therefore % must be changing
from negative to positive. Moreover, since a change
from positive to negative, or vice versa, can only occur
by passing through one of the values zero or infinity,
we must search for the maximum and minimum values
among those corresponding to the values of a given by
q' (a) = 0 or by p'(a) = 00.
163. Further, since % must be increasing when it
& *........ dºy
changes from negative to positive, dº? if not zero must
then be positive; and similarly, when % changes from
dºy Q}
da”
at another form of the criterion for maxima and minima
values, viz. that there will be a maximum or minimum
positive to negative must be negative, so we arrive
200 DIFFERENTIAL CALCULUS,
- † . . da
according as the value of a which makes º ZGI’O OI’
. . . . . d” e tº a tº º
infinite, gives º a megative or a positive sign.
164. Properties of Maxima and Minima Values.
Criteria obtained Geometrically.
The following statements will be obvious from the
figures of Arts. 159, 160.
(a) According to the definition given in Art. 161,
the term maximum value does not mean the absolutely
greatest nor minimum the absolutely least value of the
function discussed. Moreover there may be several
maafima values and several minima values of the same
function, some greater and some less than others, as
in the case of the ordinates at A, B, C, ... (Fig.,
Art. 159).
(8) Between two equal values of a function at
least one maa'imum or one minimum must lie ; for
whether the function be increasing or decreasing as
it passes the value [MP, in Fig., Art. 159] it must, if
continuous, respectively decrease or increase again at
least once before it attains its original value, and there-
fore must pass through at least one maximum or mini-
mum value in the interval.
(y) For a similar reason it is clear that between
two maxima at least one minimum must lie ; and be-
tween two minima at least one maximum must lie. In
other words, maxima and minima values must occur
alternately. Thus we have a maximum at A, a
minimum at B, a maximum at C, etc.
(6) In the immediate neighbourhood of a maximum
or minimum ordinate two contiguous ordinates are equal,
one on each side of the maximum or minimum ordi-
nate; and these may be considered as ultimately co-
incident with the maximum or minimum ordinate.
Moreover as the ordinate is ceasing to increase and
MAXIMA AND MINIMA, 201
beginning to decrease, or vice versa, its rate of variation
is itself in general an infinitesimal. This is expressed
by saying that at a maximum or minimum the function
discussed has a stationary value. This principle is of
much use in the geometrical treatment of maxima and
minima problems.
(e) At all points, such as A, B, C, D, E, ..., at
which maxima or minima ordinates occur the tangent
's parallel to one or other of the co-ordinate awes. At
points like A, B, C, D the value of % vanishes, whilst
at the cuspidal points E, F, % becomes infinite. The
positions of maxima and minima ordinates are therefore
given by the roots of the equations
ºt").
q' (a) = 00
* dy – a ... dº tº
(£) That da, T 0, or da, T oo, are not in themselves
sufficient conditions for the existence of a maximum or
minimum value is clear from observing the points G, H.
of the figure of Art. 160, at which the tangent is parallel
to one of the co-ordinate axes, but at which the ordinate
T
- _*-** * ~.
º
Wr alCilte
has not a maximum or minimum value. But in passing
a maa'imum value of the ordinate the angle aſ which the
tangent makes with OX changes from acute to obtuse,
d © . º
º , changes from positive to
negative ; while in passing a minimum value aſ changes
and therefore tan Jr, or

202 D/FFERENTIAL CA LOUI, U.S.
from obtuse to acute, and therefore dy changes from
- da:
negative to positive.
I obtuse iſ 2 acute
165. Working Rule.
We can therefore make the following rule for the
detection and discrimination of maxima and minima
values. First find % and by equating it to zero find
for what values of w it vanishes; also observe if any
values of a will make it become infinite. Then test for
each of these values whether the sign of % changes
from + to — or from — to + as a increases through that
value. If the former be the case y has a maximum
value for that value of a; ; but if the latter, a minimum.
If no change of sign take place the point is a point of
inflexion at which the tangent is parallel to one of
the co-ordinate axes; or, in some cases it may be more
convenient to discriminate by applying the test of
Art. 163. Find the sign of º corresponding to the
value of a under discussion. A positive sign indicates a
minimum value for y; a negative sign, a maa'imum.
2
When º = 0 this test fails and there is need of further
investigation*,
* See Art. 488 of the author's larger book on the subject.

EXAMPLE'S. 203
EXAMPLES.
1. Find the maximum and minimum values of y where
- 3/= (a, -1)(a) – 2)”.
Here #=(-2,+2(t–1)(x- 2)
Putting this expression =0 we obtain for the values of a which
give possible maxima or minima values
4
a;=2 and *=5.
To test these: we have
if a, be a little less than 2, do." (—) (+) = negative,
dy . . .
;-(+)(+)=positive.
Hence there is a change of sign, viz., from negative to positive as a:
passes through the value 2, and therefore a = 2 gives y a minimum
value.
if a be a little greater than 2,
Again, if a, be a little less than : } |= (–)(-) = positive,
• .. 4 dy -
and if a, be a little greater than 3’ da; = (–)(+) = negative,
dy
showing that there is a change of sign in d viz. from positive to
QC }
negative, and therefore a =: gives a maazimum value for y. This we
might have anticipated from Art. 164, (y).
dy
Otherwise: dº." (a, -2) (3a – 4),
so that when #. is put = 0 we obtain a = 2 or #.
d”y
And -- dº? T 6a: - 10,
2.
so that, when a = 2, gy = 2,
da:3
a positive quantity, showing that, when a = 2, y assumes a minimum
value, whilst, when Q} =: % º, – 2,
3 * dacº
which is negative, showing that, for this value of c, y assumes a
maximum value. *
204 JDIFFERENTIA/, OA LOUI, U.S.
dy_ 2m. 2?) -- 1
2. If - .*(*-a) (w – b)?pt 1,
where n and p are positive integers, show that was a gives neither
maximum nor minimum values of y, but that a = b gives a mini-
Illlllll.
It will be clear from this example that neither maaſima nor
e is g & e di
minima values can arise from the vanishing of such factors of .
as have even indices.
*— 7a;+6
J g
3. Show that has a maximum value when was 4
a; – 10
and a minimum when a = 16.
di
4. If % =w (*-1)*(x-3), t
show that &=0 gives a maximum value to 3/
and w-3 gives a minimum.
5. Find the maximum and minimum values of .
20:8–1532-H 36.0-i-6.
6. Show that the expression
(w--2) (a –3)”
g 7 º ºg
has a maximum value when *=3; and a minimum value when
&=3.
7. Show that the expression
a;8–33:9-1-60. H-3
has neither a maximum nor a minimum value.
8. Investigate the maximum and minimum values of the
expression -
33.3—250:8-1-60a,
9. For a certain curve
%-e- 1) (a – 2)” (a – 3)3(a) – 4)4;
discuss the character of the curve at the points
a = 1, 0}=2, & =3, a = 4.
E VAMPI, E.S. 205
10. Find the positions of the maximum and minimum ordi-
nates of the curve for which -
%– (a —2)*(2a –3)*(3a – 4)" (40 – 5)".
11. To show that a triangle of mawimum area inscribed in any
oval curve is such that the tangent at each angular point is parallel
to the opposite side.
If PQR be a maximum triangle inscribed in the oval, its vertex P
lies between the vertices L, M of two equal triangles LQR, MIQR
L
* M
R
inscribed in the oval. Now, the chord LM is parallel to QR and the
tangent at P is the limiting position of the chord LM, which proves
the proposition.
It follows that, if the oval be an ellipse, the medians of the tri-
angle are diameters of the curve, and therefore the centre of gravity
of the triangle is at the centre of the ellipse.
12. Show that the sides of a triangle of minimum area circum-
scribing any oval curve are bisected at the points of contact; and
C.

206 DIFFERENTIAL CA LOUI, U.S.
hence that, if the oval be an ellipse, the centre of gravity of such
a triangle coincides with the centre of the ellipse.
Let ABC be a triangle of minimum area circumscribing the oval.
Suppose P the point of contact of BC. Let ABC1, AB,C, be two
equal circumscribing triangles such that B, C, B2C, touch the oval at
Pi, P., on opposite sides of P and intersect in T. Then
triangle TB1.B = triangle TC,C,
Ol' #TP, . TE, sim BTE,- TG1. TC, sin CITC2.
If we bring Pi and P3 nearer and nearer to P so as to entrap the
minimum triangle, the above equation ultimately becomes
TB2=TC2;
and T being ultimately the point of contact P, the side BC is bisected
at its point of contact. The remainder of the question follows as
in Ex. 11.
EXAMPLES.
1. Find the position of the maximum and minimum ordi-
nates of the curves
(a) /=(a – 1) (a, -2) (a, -3),
(b) 4/=a:4–8w8+224%–24a,
(c) a”y=(a –ay” (w-b),
(d) afty}=(w—a)*(w-b).
2. Find the maxima and minima radii vectores of the
CUII*VéS
(a) r = a sin 6+b cos 6,
(b) reasin” 6+ b cos”6,
(6) (a^+y})*=aw”--2hay-F by”,
64 a 2 b2
(d) ŽTsingøt cos26’
0.2 b2
(e) # =l,
(f) reasinº.6 cost 6.
WXAMPLE'S. 207
3. Discuss the maxima and minima values of the following
expressions:
(a) a; (1 – ar) (1–a7),
(b) (w8–1)/(w?--3)*,
(c) sin a cos” ar,
(d) log &/w,
(e) sin” a sin mac.
4. Discuss the maxima and minima values of the following
expressions:
(a) aſſº-Fy” where aw”--2hay+by”=1,
(b) aa’--by where a y=c”,
(c) sin 6+sin q, where 6+ q = a,
(d) sin”6+sinº, where 6+5=a.
5. What fraction exceeds its pth power by the greatest
number possible
6. Divide a given number a into two parts such that the
product of the pth power of one and the gth power of the other
shall be as great as possible.
7. Given the length of an arc of a circle, find the radius
when the corresponding segment has a maximum or minimum
8.Tea,
8. In a submarine telegraph cable the speed of signalling
varies as aſſº log; where a, is the ratio of the radius of the core to
that of the covering. Show that the greatest speed is attained
when this ratio is 1: We,
9. An open tank is to be constructed with a square base
and vertical sides so as to contain a given quantity of water.
Show that the expense of lining it with lead will be least if
the depth is made half of the width, -
10. From a fixed point A on the circumference of a circle
of radius c the perpendicular AY is let fall on the tangent at P;
prove that the maximum area of the triangle APY is
**V3
208 DIFFERENTIAL CA LOUI, U.S.
11. The sum of the perimeters of a circle and a square is l.
Show that when the sum of the areas is least the side of the
Square is double the radius of the circle.
12. The sum of the surfaces of a sphere and a cube is given.
Show that when the sum of the volumes is least, eight times the
radius of the sphere is thrice the edge of the cube.
13. Show that the cone of greatest volume which can be
inscribed in a given sphere is such that three times its altitude
is twice the diameter of the sphere. Show also that this is the
cone of greatest convex surface inscribable in the sphere.
14. Find the cylinder of greatest volume which can be in-
Scribed in a given cone.
15. Show that the right circular cylinder of given surface
and maximum volume is such that its height is equal to the
diameter of its base.
16. Show that the semivertical angle of the right cone of
s - © tº 1.
given surface and maximum volume is sin 3
17. Show that a triangle of maximum perimeter inscribed
in any oval curve is such that the tangent at any angular point
makes equal angles with the sides which meet at that point.
Show also that if the oval be an ellipse, the sides of the triangle
will touch a confocal.
18. If a triangle of minimum perimeter circumscribe an
oval show that the points of contact of the sides are also the
points where they are touched by the e-circles of the triangle.
19. Show that the chord of a given curve which passes
through a given point and cuts off a maximum or minimum area
is bisected at the point.
20. Find the area of the greatest triangle which can be
inscribed in a given parabolic segment having for its base the
bounding chord of the segment.
21. In any oval curve the maximum or minimum chord
which is normal at one end is either a radius of curvature at
that end or normal at both ends.
22. Show that if a triangle of minimum area be circum-
scribed about an ellipse the normals at the points of contact
meet in a point, and find the equation of its locus.
EXAMPLES. 209
23. Find the co-ordinates of the limiting position, when
a' = a, b =b of the intersection of the straight lines
* - / * - /=
cº a-F b = 1, a' + b' l,
where aſ +b^ = d/º-H b”= 20°.
Find the co-ordinates of the point on the locus of the limiting
position of the intersection, which is at a maximum distance
• * a. e º , C
from the origin, and prove that the maximum distance is W2 º
º
[I. C. S., 1892.]
CHAPTER XIV.
UNDETERMINED FORMS.
166. ELEMENTARY methods of procedure have been
explained in the first chapter.
We propose now to show how the processes of the
Differential Calculus may be employed in the deter-
mination of the true values of functions assuming
singular forms, and shall discuss each singularity in
order (see Art. 16).
167. I. Form;.
Consider a curve passing through the origin and
defined by the equations
º
y = p (t).
Let a, y be the co-ordinates of a point P on the

y
UN DETERMINED FORMS. 211
curve very near the origin, and suppose a to be the
value of t corresponding to the origin, so that
ºb (a) = 0 and alº (a) = 0.
Then ultimately we have
Lº–Litan PON=1...}=1,...}}
*—"d, “a Jºy
nCe ‘b () ſº ºb'(t).
He € Lt-a al, (t) /*-a; (t) 3.
and if º be not of the form when t takes its assigned
value a, we therefore obtain
q, (t) ºp' (a)
*-º-º:
But, if º % be also of undetermined form, we may
repeat the process; thus
Alº" (t)
proceeding in this manner until we arrive at a fraction
such that when the value a is substituted for t its
numerator and denominator do not both vanish, and
thus obtaining an intelligible result—zero, finite, or
infinite.
Ex. 1. Lt.-,+*
Here 4 (0)=1 — cos 0 and p (0)=0°, which both vanish when 0
vanishes.
q'(0)=sin 0 and p'(0)=20,
which again both vanish when 6 vanishes.
p"(0)=cos 0 and p" (0)=2,
whence q"(0)=1 and p"(0)=2.
Therefore Lto- |-º - . tº
14—2
212 JDIFFERENTIAJ, CA LOUI, U.S.
e6+ e -6+ 2 cos 0– 4 O
4 * * * * I, - -
Ex. 2 '9–0 64 [form .
60 – e – 9 – 2 sin 9 0-
= Lto-0- 493 |form;
e6+ e-6 – 2 COS 9 0
e6 – e – 6 + 2 sin 9 0
=Lto-0–51; H [form;
_r, e^+e−"+2cos' 4–1
- ***6 = 0 24 T 24 T 6
EXAMPLES.
Find by the above method the following limits:
a’ — I & – sin a cosa.
1. Lt.-0 bºll e 6. Lt.-0− 4:3 •
sin aa, - a" – 23:8–4a:*-ī-9a: – 4
2. *-osin ba, 7. Lt.-1 a 4–2a:*-ī-2a – 1
ace” – log (1+a; e” sin a – a – a 4
a 1...”-ºltº. 8 a...rººt-º.
w cosa – log (1+a; - 3 tan & – 30: – arº
4. Lt.-, *** 99 (1+2), 9. 1.3tnº-º.
a;2 &:5
- T
Sin ( & + + ) – 1
cosh a — cos a - ( †)
5. Llº- 0 s s 10. J.t T º
Q: SII) ([? * = log sin 20

168. II. Form 0 × Co.
Let q (a) = 0 and J. (a) = 0 , so that b (a) \le (a)
takes the form 0 × 00 when a approaches and ultimately
coincides with the value a.
Then Lt.-a b (a) \, (w) = Ltº-a *.
ºr (a)
e |
and since — = 0,
QC
–––
*]: (a) T
UAWDETERMINED FORMS. 213
the limit may be supposed to take the form . and may
be treated like Form I.
6 1
Ex. 1. Lita a 6 cot 6 = Lta_0– == Lt —— = 1.
6 = 0 0+0+an 6 0–0 secº 6
. Q. . Cl
sin; sin;
. (l f
Ex. 2. It, -., a sin – = Lt, ... -- Lt, a F (0.
Q} 1. :=0 (l
169. III. Form *-.
OO
Let b (a) = 00, Jº (a) = 00 , so that ºb (a) takes the
(a)
form . when a approaches indefinitely near the
value a.
The artifice adopted in this case is to write
l
* (*) ſº
*!, (a) 1
ºb (a)
Then since wo- : = 0, and º F . = 0, we may
consider this as taking the form 0 ° and therefore we
may apply the rule of Art. 167.
o, ſº &
Lt.-, *%-Lº-LL.Bººl
alº (a) _1_ b (a)
q (a) [b (a)]
=1...}}}.}
214 DIFFERENTIAL calculus.
I’ $ (q) |f|, b (a)| * (v)
Therefore Lt.--a alº (a) |- alº º Lt.-a q' (a) '
ºb (a)
nº (a)
1–1, ºut-º.
OT $ () iſ $ (q)
Lt.-a Aſ (a) Lt.-a alº' (a) '
Hence, unless Lt.-a be zero or infinite, we have
170. If, however, Lt, a q (a) be zero, then
p(a)
q (a) + b (a)
Lt, a *-*-H+ = 1,
!, (a)
and therefore, by the former case (the limit being neither zero nor
infinite), -
—Lt...?'(*): "(*).
W (a)
Hence, subtracting unity from each side,
*(*) Lt } (*)
Lt.--a ;=nl- !/(a) '
Finally, in the case in which
Lt.-.”–2, then Lt...”-0.
p(a) 2 q, (a)
and therefore by the last case
=Lt...º.º.
*(*)'
therefore I,t $ (q) = Lt q' (a)
*-*WG) - “WG).
This result is therefore proved true in all cases.
171. If any function become infinite for any finite value of the
independent variable, them all its differential coefficients will also
UNDETERMINED FORMS, 215
become infinite for the same value. An algebraical function only
becomes infinite by the vanishing of Some factor in the denominator.
Now, the process of differentiating never removes such a factor, but
raises it to a higher power in the denominator. Hence all differential
coefficients of the given function will contain that vanishing factor
in the denominator, and will therefore become infinite when such a
value is given to the independent variable as will make that factor
vanish.
It is obvious too that the circular functions which admit of infinite
values, viz., tan ac, Cota, Seca, CoSec a, are really fractional forms,
and become infinite by the vanishing of a sine or cosine in the demo-
minator, and therefore these follow the same rule as the above.
The rule is also true for the logarithmic function log (a – a) when
I
w = a, or for the exponential function bºº when was a, b being sup.
posed greater than unity.
172. From the above remarks it will appear that if p (a) and
!, (a) become infinite so also in general will p"(a) and p'(a). Hence
f
at first sight it would appear that the formula Lt, a # is no
- q (a)
better than the original form Lt, a &
...? ()
q' (a)
p' ( w) 3 when a' = a, can be more easily
T
log 6 -:)
tan 6
But it generally happens
that the limit of the expression
evaluated.
g • *º e OO
Ex. 1. Find Ltd iſ which is of the form : .
T; CO
Following the rule of differentiating numerator for new numerator,
and denominator for new denominator, we may write the above limit
l
T
0-3
0-gº
which is still of the form • . But it can be written
CO
T
2
= Ltd – 20:2 which is of the form 0
6 =: 6 — T 0
2
– 2 cos 6 sin 6 -
_T —--~. = 0.
2
216 J)IFFERENTIAJ, (AJCUI, U.S.
acº * * * * OO
Ex. 2. Jºvaluate Lt, ..., dº ’ which is of the form -.
-- OO
acrº mº-l
Lt.-- 27 -: Lt.-- TøT F . . .
= Lt.-, ... = 1 =0.
It is obvious that the same result is true when m is fractional.
Ex. 3. Evaluate Lt.-ba.” (loga)", m and m being positive.
This is of the form 0 × 00, but may be written
* Y º
Lt.-0 º |Form ..]
a; m
º,
and by putting aº-e-v this expression is reduced to
71, º,
nº *
Lty=. − = 0 as in Ex. 2.
173. Form oo — ob.
Next, suppose p (a) = 00 and alº (a) = co, so that
q (a) — ſº (a) takes the form oo — oo, when aſ approaches
and ultimately coincides with the value a.
Let u = p(a) – ºr (a) = \le (a) ; * 1}.
From this method of writing the expression it is obvious
that unless Ltd—a q (a) = 1 the limit of u becomes
alº (a)
alº (a) × (a quantity which does not vanish);
and therefore the limit sought is oo. -
But if Lt.-a $º = 1, the problem is reduced to
the evaluation of an expression which takes the form
oo x 0, a form which has already been already discussed
(II.).
UNDETERMINED FORMS. 217
Ex. It, 6 (; -- cot •) - I.- : (1 – a cota)
= Lt.-0
a sin aſ 0
a sin a which is of the same
sin a + æ cosa; form still
sm w- a cosa. (which is of the form ...)
- Lt.-0
sin a + a cosa,
= Lt, o ºſ------0.
2 cosa; - a sin at
EXAMPLES.
1. Find It, 2, 2° sin # *
& T
2. Find Jit, 1 sec 2. ' log v.
3. Find Jt.-0 coSec” aſ — *
for the values 77 = 1, 2, 3.
4. Find Jº-0 logian a tan 2°.
2–
5. Find Jº-1 f Cot Tºp & }} e
Q? (º" — ſº
174. W. Forms 0°, co", 1°.
Let y = 'w", u and v being functions of a ; then
logo y = w logou.
Now logo1 = 0, log.oO = 00, log.0 = – o ; and therefore
when the expression w” takes one of the forms 0", co",
1°, log y takes the undetermined form 0 × 00. The
rule is therefore to take the logarithm and proceed as in
Art. 168.
Ex. 1. Find It, oaº, which takes the undetermined form 00.
1.
Lt.-0 logº-Lº-ºº- Lt.-0 +=Lt.-- (—a.)=0,
--- *
3: a;2
whence Lt.-0a:" =e"= 1.
218 DIFFERENTIAL CA LOUI, U.S.
Ex. 2. Find Lt.-g (sina)**. This takes the form 1°.
T 2
L', T (sin a)an 3– Pl, T etan & log sin *,
2 2
- e log sin a cot aſ
and Lt. It tan a log sin a = Lt. trº-H = Lt. Ir —.3.
r=; *=; cota. *=; - cosecºw
= Lt. T (-Sina; Cosa:)=0,
32 = \
2
whence required limit=e"= 1.
A slightly different arrangement of the work is exemplified here.
175. The following example is worthy of notice,
VIZ.
Lt.-a (1 + q (a)}**,
given that (b (a) = 0, \!, (a) = 00,
Lt.-a ºb(a) , (a) = m.
We can write the above in the form
1 Tºp (a). Ji (ac)
1. In tº 5
which is clearly e” by Art. 14, Chap. I.
It will be observed that many examples take this
form, such, for example, as
1.
tan w\*
Lie- (**)
on p. 9, and Exs. 21 to 26 on p. 12.
176. % of doubtful value at a Multiple Point.
The value of % takes the undetermined form O at
()
a multiple point,
EXAMPLE'S. g 219
The rule of Art. 167 may be applied to find the
true limiting values of * for such cases, but it is
da,
generally better to proceed otherwise.
If the multiple point be at the origin, the equations
of the tangents at that point can be at once written
down by inspection and the required values of % thus
found. {U
If the multiple point be not at the origin, the
equation of the curve should be transformed to parallel
axes through the multiple point and the problem is
then solved as before.
tº di tº &
Ex. Consider the value of # at the origin for the curve
a 4-H aa”y + bay”--yº–0.
The tangents at the origin are obviously
a = 0, y =0, aa. -- by = 0,
making with the axis of a; angles whose tangents are respectively
OO , 0, - b 3.
which are therefore the required values of | &
EXAMPLES.
Investigate the following limiting forms:–
log (1–a7%) 1 + COs Træ
1. Jº-0 logoosa & 4. Lt.- tanºra, &
2003 – 30:2-4-1 3:
2. Jº-1 3.05–5 tº E2. * 5. Lt.-a log (s *mº ..) Cot (a. --- a).
1 — t logs. ... COsa,
3. Lt.--º- 6. L.,"**::
ti 1–W2 sin & lo a cos;
3. 9
SlT1 —
2
~4
220 DIFFERENTIAJ, CA.I.C.U.I., U.S.
cot 6 tan-1 (m tan 6)—m cos”
7. /*a-0 *
* 6
2 *.
Slil 2
1.
8. Lt.-0 (cosa, cotºa, 9. Lt. , (1–32).98 (1-3).
0 2 = 1
10. Lt.- (loga)” “T”.
Aºn -- Bºn - 1 + Can-2-1-..
11. Lt.-2, aaºmi H-bºm-1--cam-2-1- . according as m is >, <=, or
- <772.
g sinhºa,
12. Find Lt.-0 : &
(\}* COS (º
e vers - 1 &
13. Find Lt.-0 + .
A/2a-aº
* a2+ax+a;2– A/a2-aa-Ha.2
14. Find al-, *-y +44.
Wa-Fa – Wa-a,
e & – 1 e-2
15, Find Lt.-0 24.2 + 2nsinhº'
sim 23,--a sin a
16. If a...ºne
be finite, find the value of a and the limit.

sinh 3a;+ aſ sinh 24, 4- a, sinha,
a;5
have a finite limit, find it and determine the necessary values
of ai and ag. *
17. If Jºz–0
COS4&-H ai COS23-F as
6,4
18. If Jº-0
have a finite limit, find it and determine the necessary values
of ai and ag. -
sin & +a16°--age-º-º- as log 1 + æ
a;3
19. If Lt.-0
have a finite limit, find it and determine the values of a1, a2,
03 te
EYAMPLES
221
20. Show that
21. Find
where
22. Find
23. Find
24. Find
Jº-0
J.t
l 4?
(1+xy-e-Hº
_Ile
a;2 T 24
d”y
0=0 ºz
y=6|sin 6 and 6=vers 1 a.
Jitz-
*...(
1.
sinh as Naº
0 J.' &
1
, sinh w— sy
6 =
1.
(a) Jutz-0 (cosh a); 5
I
cosha – 1) &
0 a... ſº,
2
@ 11,-, }
24 cosha – 24 — 12.3%
a;4
1.
º
CHAPTER XV.
LIMITATIONS OF TAYLOR'S THEOREM.
CONTINUITY.
177. SUPPOSE that portion of the curve y=qa, which
lies between two given ordinates
AL (w = a) and BM (a = b)
to be drawn. Then if we find that as a, increases
through some value, as ON, the ordinate ba suddenly
changes from NP to NQ (say) without going through
the intermediate values, the function is said to be dis-
continuous for the value a = 0N of the independent
variable.

O A N i. Y
Thus for a function ſpa to be continuous between
two values a and b of the independent variable, it is
necessary that its Cartesian graph y = pa shall be able
to be described by the motion of a particle travelling
CONTINUITY. 223
along it from the point (a, ba) to the point (b, pl.)
without moving off the curve.
178. In the same way, if at a point P on a curve,
if the tangent suddenly changes its inclination to the
axis of a without going through the intermediate
positions there, as shown in the accompanying figure,
there is said to be a discontinuity in the value of q'a.
Y
179. If the curve y = pa cut the w-axis at two
points A (a = a, y = 0) and B (a = b, y=0), it is obvious
that provided that the curve y = pa and the inclination
of its tangent be finite and continuous between A and
B, the tangent to the curve must be parallel to the
a!-axis at Some intermediate point P between A and B.
Y


224 DIFFER/WWTIAL CA.J.O'ULUS,
It is also evident from a figure that the tangent
may be parallel to the w-axis at more than one inter-
mediate point.
y
Ol. - BWTX
180. We thus arrive at the following important
result:
If any function of a, say ſpa, vanish when a = a and
also when w = b, and is finite and continuous, as also its
first differential coefficient ºp'a, between those values, then
q'a must vanish for at least one intermediate value.
EXAMPLES.
l
1. Show that e º takes the form 0 or oo according as a
is very small and positive, or very small and negative. Give
a graphical illustration by tracing the curve
1
3/4-1=e &
2. Show that if a rational integral function of a vanish for
m values between given limits, its first and second differential
coefficients will vanish for at least (n − 1) and (n − 2) values of a
respectively between the same limits. Illustrate these results
geometrically.
3. Prove that no more than one root of an equation f(a)=0
can lie between any adjacent two of the roots of the equation
f'(a)=0.
4. Establish the result of Art. 179 from the aspect of a
differential coefficient as a measurer of the rate of increase.
5. Show that no algebraic curve ever stops abruptly at a
point.

TA YLOR'S SERIES, 225
Taylor’s Series.
181. From the extreme generality of Taylor's Series
there is much difficulty in giving a rigorous direct
proof.
It is found best to consider what is left after taking
m terms of Taylor's series from f(a)--h). If the form of this
Temainder be such that it can be made smaller than any
assignable quantity when sufficient terms of the Series are
taken the difference between f(a + h) and Taylor's Series
for f(a + h) will be indefinitely small, and under these
circumstances we shall be able to assert the truth of the
Theorem.
182. Lagrange-Formula for the remainder
after the first n terms have been taken from
Taylor's Series.
THEOREM.–If f(a + 2) and all its differential coeffi-
cients, viz. f (a + 2), f' (a + 2), ...... f*(a + 2), up to
the m" inclusive be finite and continuous between the
values 2 = 0 and 2 = h of the variable increment 2 then
will
An-1
h? ... . - º,
ſº-ſºº wºr"()+...+ºf+. ()+; freton)
where 6 is some positive proper fraction.
Let
h
7, 2 **-l ??,
f(a + h)= f(a) + hf'(a) +: "(...) +...+(−iſiſ"(*) +. R... (1),
R being some function of a, and h, whose form remains
to be discovered.
E. D. C. 15
226 DIFFERENTIAL CALCULUS,
Consider the function
2 ºv--1 ??,
f(a +z) = f(a) – 2f'(a) -ire T • , s -ºf" ()–: R=q, (2), say;
them differentiating m times with regard to 2 (keeping
a constant), -
m-2 7, 1
ſº - f'()- "@-...-ºf”()-tºº-ºº.
º-3 ºv-2
f" (a + 2) -- ſ"()-...-Hºº-ºo-ºº-ºº.
etc., etc., etc.
f*-* (a + 2) - f*-1 (a)- 2R=q{*}(z),
f*(a + 2) - R=q,”(z).
All the functions ºp (2), p'(z)..., dº (2) are finite and
continuous between the values 0 and h of the variable 2,
and evidently p(0), q' (0), p"(0)..., p"(0) are all zero.
Also from equation (1) © (h) = 0. Therefore by Art. 180,
$' (z)=0 for some value (h) of 2 between 0 and h,
$" (z)=0 for some value (h) of 2 between 0 and h,
... b”(2) = 0 for some value (hº) of 2 between 0 and h,
and so on; and finally
q,” (2) = 0 for some value (ho) of 2 between 0 and h, ı.
Thus f" (a + hn) – R = 0.
Now since ha < ho_1 < ho_2 ... <h, ºh, k h,
we may put ha = 0h where 6 is some positive proper
fraction.
Thus R = f°(a +01).
Hence substituting in equation (1)
2 72-1 Hºt
fern)=f(x)+\r()+; rºt. +ºf+0+ if"(.401)......(2).
TA YLOR'S SERIES. 227
This method of establishing the result is a modifica-
tion of one due to Mr Homersham Cox (Camb, and
Dublin Math. Journal).
183. If then the form of the function f(a) be such
that by making n sufficiently great the expression
h?
iſ” (a + 6h)
can be made less than any assignable quantity however
small, we can make the true series for f(a + h) differ by
as little as we please from Taylor's form
for for; fºr ... to 2.
The above form of the remainder is due to Lagrange,
and the investigation is spoken of as Lagrange's Theorem
on the Limits of Taylor's Theorem.
184. The corresponding Lagrange-formula for the
remainder after n terms of Maclaurin's Series is ob-
tained by writing 0 for w and a for h and becomes
a;? 70, - t
thus giving
,2 ,??-1 •??,
re-ſo) ºr() ºr 0+...+ºf+0+:/º.
185. The student should notice the special cases of
equation (2), Art. 182, when m = 1, 2, 3, etc., viz.
f(a + h) = f(a) + hſ" (a + 6,h),
f(a + h) = f(a) + hy’ (a) + ºf (a + 6,h),
etc.;
all that is known with respect to the 6 in each case
being that it is a positive proper fraction. - *
15–2
228 DIFFERENTIAL CALOUI, U.S.
186. Geometrical Illustration.
It is easy to give a geometrical illustration of the
equation
f(a + h) = f(a) + hy" (a + 6h).
For let a, f(a), be the co-ordinates of a point P on the
curve y=f(a), and let a +h, f(a +h) be the co-ordinates
of another point Q, also on the curve. And suppose
the curve and the inclination of the tangent to the
curve to the axis of a to be continuous and finite
between P and Q; draw PM, QN perpendicular to OX
and PL perpendicular to QN, then
f(a + h)-f(a) NQ – MP LQ
h † = }=tan LPQ.
*

ö-f M S N X.
Also, a + 6h is the abscissa of some point R on the
curve between P and Q, and f' (a + 6h) is the tangent
of the angle which the tangent line to the curve at R
makes with the axis of w. Hence the assertion that
ſº-ſº-ſº on
is equivalent to the obvious geometrical fact that there
nust be a point R. Somewhere between P and Q at which
the tangent to the curve is parallel to the chord PQ.
TAYLOR'S SERIES. 229
187. Failure of Taylor's Theorem.
The cases in which Taylor's Theorem is said to fail
are those in which it happens
(1) That f(a), or one of its differential coefficients,
becomes infinite between the values of the
variable considered ;
(2) Or that f(a), or one of its differential co-
efficients, becomes discontinuous between
the same values;
- º, -
(3) Or that the remainder º f" (a + 6h), cannot
be made to vanish in the limit when m is
taken sufficiently large, so that the series
does not approach a finite limit.
Ex. If f(a)=N/a,
f(t+h)= x/a+h, ſº-sºº.
Hence Taylor's Theorem gives
— 1
f(a + h)= Jºh-ºgºhº * Q &
1.
If, however, we put a = 0, 5–7— becomes infinite, while Vººrh
’ 2 vſa,
becomes Vh.
Thus, as we might expect, we fail at the second term to expand
Wh in a series of integral powers of h.
188. In Art. 70 the proof of Taylor's Theorem is
not general, the assumption being made that a con-
vergent expansion in ascending positive integral powers
of w is possible. The above article shews when this
assumption is legitimate.
For any continuous function f(a) in which the (p+1)"
differential coefficient is the first to become infinite or
discontinuous between the values a, and a + h of the
variable, the theorem - .
f(a + h) = f(a) + h;f'(a) +... + ºf (a + 6h),
230 DIFFERENTIAL CA LOUI, U.S.
which involves no differential coefficients of higher
order than the p", is rigorously true, although Taylor's
Theorem,
- y h.p hºp+, »
fails to furnish us with an intelligible result.
189. The remarks made with respect to the failure
of Taylor's Theorem obviously also apply to the par-
ticular form of it, Maclaurin's Theorem, so that Mac-
laurin's Theorem is said to fail when any of the
expressions f(0), f' (0), f"(0), ... become infinite, or if
there be a discontinuity in the function or any of its
differential coefficients as a passes through the value
zero and increases to the value aſ, or if the remainder
70,
º f”(6a) does not become infinitely small when n becomes
infinitely large, for in this case the series is divergent
and does not tend to any finite limit.
EXAMPLES.
1. Prove that
* = 1 l a;% l 2 &n-1 l 70–1
a”= 1 +& oga-Fºi ( ogea) +…+(n-1) (logs a)
£78
+. a?" (logº a)”.
[Here f(a) = a \, f^(a)=a^ (log, a)”, f"(0)=(logo a)”,
Mºſt wº
and m! f*(6a) = ja” (log, a)".
Hence the result follows by Art. 184.
The student should notice that
w” a” (log, a)”
7,
FXAMPLE'S. 231
can be made smaller than any assignable quantity by sufficiently
increasing n. Hence the remainder after n terms of Maclaurin's
expansion have been taken ultimately vanishes when n is taken
very large. Therefore Maclaurin's Theorem is applicable and
gives
2
wº-1+w log. a+. (log, a)*-i-... + to Co.]
2. Shew that
o:34:3 aftº:5 a "wº . nºr
3 + 5 T- is tº ſº ++ SIll 2. + ...,
and that the remainder after r terms may be expressed as
a'a' in (age--"
7" Sl 0.60% + 2 / .
3. Shew that the remainder after r terms of e^* cos by have
been taken is
sin aa-aw —
* , sº
(ºr area” cos (ºr tan-1 #) e
4. Shew for what values of a, and at what differential coeffi-
cient Taylor's Theorem will fail if
_{*-a)"(*-bº ("-o)".
f(a) (c-d);
5. How does Maclaurin's Theorem fail to expand
--- _1
log w, tan-i Wa, or eTz
I
in ascending positive integral powers of a 7 Is e º continuous
as a passes through Zero!
MISCELLANEOUS EXAMPLES.
1. A circle is drawn with its centre on a given parabola and
touching its axis, shew that if the point of contact recede
with a constant velocity from the vertex, the rate of increase of
the area of the circle is also constant.
2. If V be the volume of a sphere of radius r and S its sur-
face, shew that
G. V
;=S,
and interpret this equation geometrically.
3. If V be the volume of a cube whose edge is w, and S its
surface, interpret geometrically the equations
G. V. dS
- = 30%: ºº = 12a.
da: 30%; da, 12a:
4. If V be the volume of a rectangular parallelepiped of
edges w, y, z interpret the equations
9 V Ø V 9 V
º, =4/3, Oy = 24), 32. = @y.
5. Evaluate the limits:
I\zº
(1) Ltº-2 (, tan !) g
(2) Lt.-co 2°tan #:
d /aa”--ba;+ c
@ 11,-4(*#).
cos” tra; tr.”
6. Prove that *=}; i. -- 2e *
MISOELLANEOUS EXAMPLE'S. 233,
tan (a +a;) – tan (a – ar)
7. Evaluate Lt.-0 tan-" (a +a)-tan-" (a – wy’
/ 1 1. T
8. Prove that Liz 0 (* T2atana,) = 6.
9. Find the limiting values of the following functions
(i) a sin 6–sin a 6
6 (cos 0–cosač) when 6= 0.
a sina – when we!.
(ii) — 2
\ COS 32
(iii) (sina)** when a =0.
Wa-Wa-i-War-a
Wº a 2
(iv) when v = a.
10. Find the intercept made by the curve y=a cot . (The
Quadratrix) upon the y-axis.
11. If u=w Wºº-Hoº--a” log (w-H Wºº-Faż),
du
** = 2 A/r2 Loº
shew that da, 2 A/2-Eaſ.
------ 2 — or?
12. If ºwº-alogº
y
shew that #– TV #.
& d 1, 1, 1, 1
13. Find do | 1, w, 1, 1
1, 1, 3, 1
1, 1, l, w
14. If 3/= 30, Q, b, 0 3
—a, w, - c, b
– b, c, a, – a
– c. – b, a, w
shew that dºy 12a:}=4 (a4+b^+ c”)
da;2 *. e
234 DIFFERENTIAL CALOUI, U.S.
15. If VT-º-H WI-32=a (a –y),
prove that dy F WI–ſº o
dw WI-32
I 1+w W2+wº I ..., wV2
= -—- sºme-mº- − tº ! ---, ,
16. If y 4 M2 lo invisiº" 3.3 * “T.
dy l -
prove that da, TT-Eaſ'
Hence expand y in powers of w.
17. If
OY, 0.2% asſº
= 1 + → *—+ + ...
*T*"º-ſa," (n-aja-ay" (ºta)(ria)(`Ea.)
to n +1 terms,
dy y 0.1 Q2 0,,
shew that #=}|ºtºr...+º,
– T \4
18. Differentiate ğ. twice with respect to w, and find
the value of the result when was 0.
19. If u, = cosh”a, and va-sinhºa, prove that
dºw,
º: =nºu, - n (n − 1) un-2,
d”v,
and dº? =nºva-Fr. (n-1) vo–9.
20. If w=cos logy, shew that (1–a7%) y2–ay1=y.
21. If 3/=(aa’--ba-i-c)/(1-3),
shew that (1-0)3/3=3%.
* Gºt d º,
Prov - - - – ø0% ——
22. Prove that dºw” (e”. w)=e (#6) 20.
riod 2"/ -- I
23. Find dº where y Ta;2+6, +8°
d” paº-Egº-Fr
da" (a –a)(a) – b)(a) – c)
ems 72, pa”--qa-Hr l
-(-) m. I 2.É.-jº (Tºri.
24. Shew that
24 ap ap
* F + ſū’ — ---
O="j (I outſ;a p} p
qbul Aous ‘uſ I-24) º F="aſ JI oolio H
20,
'0–"ſ; (1 +w) w — "t "ſivg-Hºt “ſ (I — ga) (q)
‘ſawz=% (I —gº) (p)
oAOId ‘iſ – gºr)=ſ, JI “Ig
.6 x_3=º (, – I — w8%0 wºp
fº = /*-up (I —w)?t-i- Žºp &ltz+ ſºup gº
‘Joãoqui Moaa Aſue aq at J put
§ ap gºo gº
{ =ſ. m-º-º-º-º: ----- -: tº
*-******* usin -ia-6 II oc
ſaro-FQ zºo-HaqZ + 0
{º} 1-U84 (i+1)} SOO º, (errºr) itſ w(I–)=
T-Fº
zºo-Haq6+p ºp
a’9-H Q wºo
‘..Q.<00 JI qetſ, oAOld '6%
‘(,0–I)” “(40 – I) (2 – I)/ w =(0).tºp
qeq Aous ‘Āqun uſeuq SSol SI o
oloux ‘oo on “1-(ago – I)1–(wo – I)1–(a – I)=(a) ºf JI '8%
. ſtru(***) (, —
{ “Ow *(I–)--
... truſ&#8) *º-ºº: (I–)=
*Ow '04 'O. i 20, ºl,
1-(?!-Ha!)'''1-(3+4)1-(I+2)1–0, º qbū, AbuS 12,
º
.1+ w(wć– I) 3. gº u(wć-I) 9. ., grºë-I) 6–
i u z-ag (I–w) u " i (i+u) * * i (3+w) w8
g(wć - I) (...) g
(º-FT) + I) w\ p qbul AoûS '9%
wº
º; pug ‘a uſs wºr=ſº JI g3
98% 'S'ſ Toſſ WYMT Sſ) OſſWWTTGIOSIW
236 DIFFERENTIAL CA LOUI, U.S.
32. Prove that ea + ba + c2+ da:34 . . .
2 #3
= 6% {+*** * ******, ~).
33. If u = f(a), shew that
& a du , 1 /a:\” d”u 1 /w’N3 dºw -
y()-u- a. * * (..) dº? T 3. () ... +…
34. If e” sin a =>anº”, shew that -
. 707ſ.
0. O!, 0n-3 sn-g
* – 1 * -- 2 * — ;
al-HP+-gº-ºº--...-- [I. C. S.]
35. If e”=log (a6+a1,4-a,”+...+a,”+...), prove that
0n-1
1 !
+*****4...+.
o 70,
(n + 1) an 11=a, + 3 |
36. If Aa, A1, etc. be the successive coefficients in the ex-
pansion of y=e^*****, prove
72 7777.
7)?) rar . rit -
4n+1=ni {4. +: i4n-r (co- g-sin #) . [I. C. S.]
37. Given that
A. A 43
sin log(1+x)= # *4; *+. a"--...
B B. B
cos log(1+x)= 1 +: wiłł ***** tº e ∈
calculate the first eight coefficients of each expansion.
[M. TRIPOS.]
38. From the expansion of sin-la/VI-.”, deduce
tan-l ––– 1+*, *, +*.* w” "...
*T**II. 'Tá Hºt 3.5 (IEA) + “j.
Also establish the series
1 1. 2 1. 2. 3
7t
(a) ;=1+3+...++.
. 5
2 1 / 1\ 1.2 /1\? I . 2. 3 /1\3
(b) ja-l +}()+;() * 3. 5.7 () + . . . . . .
MISOELLANEO US EXAMPLE'S. 237
39. Establish the expansions
Tr? 1 1 , 1 1. 2 . 1 1. 2. 3
(9) ā-143: 3+3. 3.5+iāº; #~
T2 I 1 / 1 I 1.2 /1\? I 1.2 . 3 /1\3
(b) # = i+}()+...; (...) +}#}() + ......
4tr 3 3.6 3.. 6.. 9
40. Prove that sº-1+...+º, + digit.….
41. Shew that if f(a + h) be expanded by Taylor's Theorem
and then h be put equal to — ar, the sum of the first n + 1 terms
may be expressed as
(-1)^* * [ſº]
7, da” (º
3 a.3
42. In the curve : +% =ay, find the points at which the
tangent is parallel to one of the co-ordinate axes.
43. Find at what angle the circle aº-H4/*= a (v-y) cuts the
co-ordinate axes.
44. In the curve y=log coth; shew that #: =coth ar.
d
45. In any curve prove that
7,206
(a) p-º.
Tº "dº
(b) W r"—p ds
46. Find the sine of the angle of intersection of the rect-
angular hyperbola wº–y”=a” and the circle w”--y?= 4a2.
- 2
47. Shew that the points of inflexion on the cubic y= i.
are given by a =0 and w= + a v3. -
Shew that these three points of inflexion lie on the straight
line a =4y.
48. If a line be drawn through any point of a given curve
at right angles to the radius vector (and therefore touching the
first negative pedal), then the portion of it intercepted between
the two curves is equal to the polar subnormal of the point on
the original curve through which it is drawn.
What is the geometrical meaning of %
238 DIFFERENTIAL CA LOUI, U.S.
49. In any curve the radius of curvature of the evolute at a
point corresponding to the point p, r on the original curve is
−, d / dr
2– 2 --- * —
W, *#(º).
50. If p and p' be the radii of curvature at corresponding
points of a curve and its evolute, and p, q, r are the first, second
and third differential coefficients of 3/ with respect to a, prove
that
p'ſp={3pg” – r (1+p”)}|q”.
51. If p, p' be the radii of curvature at the extremities of
two conjugate diameters of an ellipse, then
(p34-pº) (ab);=a^+5%
52. The projections on the v-axis of the radii of curvature
at corresponding points of y=log see w and its evolute are equal.
53. In the curve for which
dy).”
27), [ . . — rv2m 2n
Q/4” -i- ) = 0.47° – ?
3/ (# 3/*,
prove that the normal is m times the radius of curvature.
54. Shew that there is an infinite series of parallel asymptotes
to the curve (r-b)6- a cosec 6, whose distances from the pole
are in Harmonical Progression. Find also the circular asymptote.
55. Shew that the asymptotes of the curve
4 (cº-i-y”) – 174%–4a (4/*-a”)+2 (wº–2)=0
cut the curve in eight points lying upon an ellipse whose eccen-
tricity = x/3/2.
,2-72 --
56. In the curve a cos vº. b 6=W/a2–bº,
C!?”
W52-ETA
57. Find the asymptotes of the curves
shew that p=
(1) wº—yº-a”ay.
(2) (a"—2aw) (v*-Hy”)=b2a9.
MISOELLANEOUS EYAMPLE'S. 239
58. Find the asymptotes of the curve
(w–3)*(w-29) (w-38)—2a (w8–y”) —2a” (a +y) (c.— 2/)=0.
59. Determine from the equation
(a – a "+7 (y–b)*=(w-i-y–a–b)",
dy
the values of . when a = a, y =b.
da,
60. Find that point on the curve
3/\" aſ
b / Tº a
where the angle which the tangent makes with a radius vector
from the origin has a maximum or minimum value.
61. Find the area and position of the maximum triangle
having a given angle which can be inscribed in a given circle
and prove that the area cannot have a minimum value.
62. If four straight rods be freely hinged at their extremities,
the greatest quadrilateral they can form is inscribable in a circle.
63. Find the triangle of minimum area which can be de-
scribed about a given ellipse, having a side parallel to the major
axis of the ellipse.
Also shew that the triangle formed by joining the points of
contact is an inscribed triangle of maximum area.
64. A tree in the form of a frustum of a cone is n feet long
and the greater and less diameters are a and b feet respectively,
shew that the greatest beam of square section that can be cut
720,
3 (a – b)
out of it is feet long.
65. Find the maximum radius vector of the spiral
7' cosh 6 = a.
66. Investigate the maximum value of cosma. cos”.
67. Investigate the maxima and minima of
cosa;+COS 23-F coS 33.
68. Find the maximum and minimum values of e” cosa; and
trace the curve y=e” cosa.
240 DIFFERENTIAL CALCULUS,
69. The corner of a leaf is turned down, so as just to reach
the other edge of the page; find when the length of the crease
is a minimum; also when the part turned down is a minimum.
70. If the angle C of a triangle ABC be acute and constant,
prove that sin.” A + sin” B is a maximum and cos”A +cos” B a
minimum, when A = B.
71. Shew that the shortest normal chord of the parabola
3/*=4aa is 6a M3 and its inclination to the axis is tan-1 M2.
72. Find the maximum value of
(1) when a-b,
(w—a)*(a –b), ſº when a <b.
What happens if a =b Illustrate your answers by diagrams
of the curve
3/= (a, - a)*(w-b)
in the three different cases. [I. C. S.]
73. Prove that the pedal equation of the envelope of the
line a cos 20+y sin 20=2a cos 0 is pº- (r”—a”).
74. A square slides with two of its adjacent sides passing
through two fixed points. Shew that its remaining sides touch
a pair of fixed circles, one diagonal passes through a fixed point,
and that the envelope of the other is a circle.
75. The tangent at any point P of a parabola meets the
tangent at the vertex in Q, and the normal at P meets the axis
in R; find the envelope of QR,
76. Ellipses of given eccentricity e, have for their major
axes parallel chords of a circle of radius c. Shew that their
envelope is an ellipse whose eccentricity is w/#. ſº
77. Shew that the expansion of s º is
1+ º- nºt P.;_ Bºr e e e in a º
where Pi—, *-*. *-*. *-*. etc.
[These numbers are called Bernoulli's numbers.]
MISOELLA WEOUS EXAMPLES. 241
78. Shew that if
J a;3 4:5
tan was Si T + Sº 3] + Sº 51* tº e º º is e
22m (22m — I
then Sºn—1=Ben-1 *=p.
79. Shew that
a;2 wº a;6
Seca'-1+S, 2. + S, 4. + S6 6t tº º e º e sº
where S2= 1, S1=5, S6–61, etc.
[These numbers are called Euler's numbers.]
J; 4:2 w8
80. If seco-tanw-1+S, iſ 4. S.; +&#14
© tº º $ tº º
prove
– 1
(a) Sp41=S, +4. º 2 's-is.
p (p − 1) (p-2) (p −3)
+ 2. 3.4 Sp
l e –4S4+ tº e º 'º e º e
(which when p is even expresses any Bernoulli's number in terms
of lower order Euler's numbers).
() s, zººs, +2(e–Pºtº
1 .. 2 1. 2. 3.4 Sp-a-......
Pſ sin PT
+cos’s sn-3.
S =272, SS-1385, SA-7936, etc.
2p**p! I\p + 1 I\p + 1 I\p + 1
@ s-ºſ()"+(−)"+(;)
I\p + 1 I\p + 1
+(−}) +(...) + ...... }.
16
242 DIFFEREVTIAL CALCULUS,
(e) Deduce the several results
1– +: -; + e tº e º & e =#;
1+,+,+,+ • * e o e e –
1-###-#4 e e s tº tº tº -:
1+,+,+,+ tº e º e s e -:
1-#4 -} + • e º & º º -;
•
etc.
11.
16.
21.
:
ANSWERS TO THE EXAMPLES.
CHAPTER I.
PAGE 5.
1 ... b ..., a
(a) o, (b) a ’ (c) o . 2. (i) a' (ii) :
3a*, 4a3, 5aš/2. 4. i. #. 5. – 1.
3 : 1
– o , 0, 0, -1, - Co., oo, 0, co. 8 2 ” 2. '
PAGE 11.
3 T}l, l
| & 2. = . , - . • T a # * *
Ogba 5 3 7, 4. 70, 5 4
- 1 3
4. 7. 2. 8. 1. 9. 2 * 10. 2. '
2 1. 1
3 12. 6 * 13. 1 14. 1. 15. 15
- 11 17 13 18 2 19 1 20. 1
6 & & 60 w * 3 & g 2 * e gº
o. 22. 1. 23. evº. 24. 0. 25. e-l, 26. e
CHAPTER II.
PAGE 16.
Y – y=3aº (X-ax). 2. Y— y = 4a:” (X - a).
2 Ma (Y-y)=X-w. 4. Y-y=(2a, +3a.”)(X—a;).
Y-y=cosa (X– a.). 6. Y— y =e" (X-ax).
a (Y-y) = X – a. 8. Y-y=sec’a (X-a).
Xa, + Yy= c”. 10. Xala”-- Yyſb°=1.
#.
16–2
244
DIFFERENTIAL CALCULUS.
l.
10.
l3.
PAGE 17.
2. 2. 1. 3. 3. 4. 6a. 5. – 1/aº.
– 1/aº. 7. – 2/aº. 8. a/2 Ma. 9. al Maº Faº.
evz/2 /a. 11. esinº cosa. 12. tan a .
sin a + a cosa. 14. (a cosa – sin a)/acº. 15. aº (log a + 1).
CHAPTER III.
PAGE 27.
aº (3 sin a + a cosa); (3 + a) aºe”; aº (3logea + 1);
aº (3 tan a + a secºa); aº (3 logo sin a + a cota).
aº (4 – a cota) coseca; (a cosa – 4 sina)la”; (cosa – sin a)/eº;
eº (sina – cosa) cosecºa.
s 1 s
secº a loge sin a + 1; eº (lose ri) ; sin a (2+tanº a).
aºeº sin a (3+ a + a cota); tan a logea + a secº a logea + tana.
aºe º (3 – a) sin a +a cosa); aºe º cosec a (3 – a – a cota);
– (3 + a + a cota)lacºeº sina.
(sin a +2a cosa)|N/a; 3 (2a secºa-tan a)/2ai; 2eº (2a – 1)ſai.
eº (2ai +- 6ai +2a + 1) /2 è ; eºlogea (aº + 5aº +3aº) + eº (aiº +aº).
PAGE 30.
cosa . 1 º
2 /sina' 2a Mog, e'
secºa . cos Ma
2 Vtana' 2 Ma
- s - va a r
esinº cosa ; etanº secºa ; 3aºeº; e /2N/a; 1.
3e3º; – e-º; 3 sinº a cosa ;
cot a ; 2 cosec 2a ; i; 3.
29; ” a
cos (log a) . secº (logga). cos (logea)
iC 3 3C 2a N/sin (loga)'
cosA/a . cot Ma
4 Va Vsim Va 2 Ma
N/eº
- - 5 A/o 1
Cot s/ eº; secºloge sin eva . cot eva evo » – ,
4 2 Ma
w/ 2
e : t—wº wº, Soo w : a Soo a t-ºuts u : aru Soo w agsoo &
Q} /* SO9
'gg HòV.I
Ø
'ao 3ol : I+ aſ : 20 30I ' 250120
g(a 301) aſ , a, , 29 + æ
**---------- ... — 9 -
9 + 3:
‘(T – a 3 * e
(I — a 3oE ar) T 20 + I
Q:
'ol : 3’āo rò 29 ºp (p 30I +I) : (u + æ) wor-ºw
: [**10+2) wo 29
'[...(a + p)], b t—w(w-H p) w : (a + m) jºb t—w[(w + p) (b] u
(a) b/w Z Q}
: =#v=º- : (a 3ol), @ 1 : (20), go
º; ; (2009; tº
. p"BoI2 • * - I , gº! — T , 9 + æq + zap , q + æp , p +a; , º&
g(v*āo) a # 6 q + æp3 p I I
'ºgog a U.Soo { a, UUIs : *wou : a 2- : &#23
‘F8 Hºvd
49b–od+, 0–ay, tº
{p5 – qd-i- a (b º
: {pb + qd-Ha (b+d)} t_a(q+a;) t—a (p + æ)
** & *(p + 2:Q -- zºo)
{q +a; (2 + p) Z +ga:Q} (9 – p)
ºv) (w-)gſvg (a + i) {(n-p)lo (A -º/ö)º g/w/ 2
‘g
; :(w/ - pſ') w/S//* : */ ogiſ(x^ "II+vg)–
… +!? 4. [** – a 4.1
tº gº gº
‘up : 1 u(ſ) + æ) pit 1–wa'apu : 1-ºwnu : 1-uſ0+ ap) pu
; (1) + (1, º Z, 0.
.# + "g .aſp-Ha). . ; &
I I T --
: t—lºw : t—w(p + æ) w
gº 8 a.0%
# ** – : a "- :*— ; : * : ºr 6
§ - g 1 t g T I T 9
frt_a:0I – ‘g a' - $ 620L ‘I
"Zg Ił)V.I
AI &I(HALCHWHO
Q#Z 'S'ſ Tcl/WP V4ſ (ſ/III, OJ, SºHº AA SAW V
‘9
“t,
‘9.
246 IDIFFERENTIAL 0AL0'ULUS,
- «- cos a:6V5inas
cot ,/a: : cos aceººº : --¡E
N.: * 2,V3in *
COS Af 3' 1
2. –ººº - COt a' ;
4 Javsin va: 2 Ja;
{w?r?-1
« → es SIntrºTi a: e
3. sinº-l a cosº-la: (m cosº a: - n simº a.) ; §t¡±(m cosºas +nsinºa.);
GOS??t=1 a:
mºaº-l sinº-1 (na”) cos (naº); eºa (a sin ba + b cos bar).
4. # (cos 2a + 2 cos 43 - 3 cos6a); sin a (sin 3a: + 3 sin a')|sinº 3a.
5. -¡ein 2a: + 2 sin 4.x + 3 sin 6a;) ;
- cosº aa . cosº ba.. cosº ca' (ap tan aa + bq tan ba + cr tan car).
PAGE 38.
e… 2
1. 2.x sec aº. tan acº; —¡— . ; 2:e secº acº;
a: J3Ti
2.c . 2a: sin a:3: 2
1-1-a:4 ? {U ' V2 = Ž '
2 —'— eº secº eº ; 2 cosec 2a: : —'—. - 2 GoSec 2a;
” 2 cosh a: * 9 * (1+ aº) tan-1 e º t.
3 1 0 1 . _ a _. 2
- ./2aa'??' ./a (2-a)-E2=(IEA) ER' aº+a** 1 + cº
· 4. - - cos ac/2 Jeovers ::: pqa'-' tan p-1 a 7. secº aº ;
---1 -1 aº p\q-1 2p\ . - a + →
pqa:p-l (tan-1 acºy-1/(1+ aºrº); log tan **(TEPHRIFTE'
tan a: 1
5. secºa·.sinTº a: +-¡-----; -; ===== Ž
JTF??' sin a: </siñº? E gosº?
5in *__.. … (..;,,-, I.
1 3.3 e S1InT! a: -H- -- T .
-i- GO8º a: ./Tayº
PAGE 42 (first set).
- sin a:
1. a: Sim 30 (e -H- COS :: .losa) 3
e-, d… :?:
(sin-1)* (log sinT! c +--------;
sin-1 aº . </1 - arº
) ; caºti log ea?; 2a:ºlog esc.
ANSWERS TO THE EXAMPLES. 247
11.
* -ºs. –02 log º ſºloo &
a 2/2–1" 5. **:/elow; 6.
* cos” a . &
(sin wycosa gin. -sinº log sin a
w;
in r sin” aſ
+ (cosa)sinº (cos a log cosa –
• COS (Ö.
- 1.
(tana)" (log tan a +2a: cosec 2a) + ætamº (* a log a + a. tan •) {}
- 1 1
Q: •3; -º-º-º-º- ------
tan aſ . loga;. e”. a.”. J:ſº cosec 2a: +++2+log+}).
PAGE 42 (second set).
3 2. -- 3. * + * --
1 + æ2 II º' 2 1 + æ2 © II.2"
2 (C 2
1. 6 —- . 7. —H·- . 8. — -; e.
M1-ſcº Ji-Hº 1 + æ%
22–93 – 2,3 2 2 – 1
*— 10 ºt. ii. H. 12. #.
MIaş 2a:# (1+a) (1+a;”) MI-2; 40° —
PAGE 46.
3:2 agn-1 º/
Ty3. 2. jºi. 3. a (; – 1) '
y 2a:” – 1 ga:9-'+ yºlogy
Tryº-Hºrºlogy'
PAGE 47.
1–7. 1–7.
a;2 + 2a: -- 2 7 1. & n 2a: 2 2 Tº,"
Tú. II).T. 2. ; (a+1) tº 3. ... (a ** e
1 5. – 3a; +a;3 6 a:4–2a2a2+ 4a:4
(1 - a)} (1 +z); (1 + *}} g (tº - a'); (v2 - 4a):
2+2a – a 3 2 (1 – wº)
--- 1. 3 * * I.I.
2 (1 - a)2 (1+a;+2%);
–– 10. <!-cosa”
a {1+(loga)*} 180° “
in oa, ****º- -1 23;
cos e. e. logº +**. 12, 198 Vcota, 2tan-le
cosh ac Sin 20,
248 DIFFERENTIAL CA LOUI, U.S.
13. simha, , i.e. tanh ar. 14 –– 2 cosec 2a: º
cosha: M2 log cotº-(log cota)?
COS (C l 1
15. —--. 16. --—a . ... — — .
1+ sin” aſ 1 + q2 e17 a Væº-1
in-l or Ym—l (aacs-1 o’ \'-1
18. (sin-la) *(cos a) (m cos-"a – n sin-1 a.).
W 1 – arº
1. ,
19. cos (e” loga) e” log (ve”) VT (log a)”
l
log as
– sin (ew log ac) —Hº-.
( g 'Jºy
/~\w 2a: \* 1
20. (...) |. (og #) + } e
70, 70, Q.
21. ab Q} 2 |an- ; + #! e
a2+ acº (an- ..) -

cos- a - a VI-.” 23. “ sin (a cosec" a.) Māº
22. º —------'. 24. — e
(1-z)? a VºII b -- a COS a.
l 3
25, 26 tan T'a, º + 3a;” tan *} º
26. eaz (a cos (b tant" a.) — b sin (b tan-ha)} .
º 1 + æ2
an. *a*{2+ºloged) as "logaesin (loga Jaºra.”)
e 1 + aya?cz © (a^+ wº) cos” (log, N/a2+2)
29 2 - 30. —; 1 -. 31 1
1 – a 4" a log a a log a logº a logº a ... log” a
32. –– . 33. += * 1 . 34, 10°. 101*(loge 10)”.
a + b COS a. Vi -- a 2 2 Jay-º
- l
35. €2. e6°. 33. eº.aº. log (ea). 37. a.e. e” log (ve).
1 o
38, a 2°. a 2. {log a)?-- log2+} e 39. a.” log (ea) – a " logí.
40. – (cota) cotºcosec” a log (ecota) + (cosh a)** sinh a log(e cosha).
sin & -
aft acca: Bin &
log \a'a cos we “
41. 1...} 1 + q2c2a:28in ×
3
• 1 – 2a:3
+ tan-1 (acºasinº) - **- &
2c% (1 + æ)?
A WS WERS TO THE EXAMPLE'S.
249
42.
43.
45.
46.
47.
48.
49.
51.
54.
57.
59.
61.
64.
65.
67.
aw-H hy 52 70 (l, E
ha, + by 2a: \a + bac" )
a; (3+2 tan log a + tan” log ac). 55.
1 e tan T 1 a.
MIET: 1 + æ2

... ??? Tºl, ... ?)!, 7??,
Sln — -- COS — 1 — sin - + cos
Q, Q} Q}
MT-3–2 vſa.
(C
2 V(º) (1-inº)
Q} Q}
770,
$)
(C-
4 Va. VI-.” VJºcosº (1+ Vº 4 cos- º'
1+ \ſº
sin e2%
º/ [* cos e” log
– y cota (1 + 2 cosec” a log cosa).
— Q log cot 1 a 1
!/ a;2 + . (1+a;?) cot-' aſ
1+2.J., 2 Jr (1+ Ja) ºl
I
1\% a +1 1 #T'ſ, *
(1+.) |log --- #|| + 0. (a + 1 – loga.).
l, a " + y” – ay
-- e 50.
(l, (v*4-y”) secº g – ba;
1
Aſ log y 1 + a log a logy
a loga; 1 – a log y 58.
_aw-H hy--g tº 60
ha;+ by +f e
6aº (1 + y”) tan aftetan", 63.
1+y” — log sec” whetan" / "
– (1+ a” cos” ba) (cº-aa. 4-a”)”-l
Q}
2
acsimTºw (og a; + MF * sin-1 •) tº
1 JTFºr MI-24
* V1+2 – VI-.”
|n (2a, + a) log cot
68. — .
QC
cosa: . cos 2a: . cos” y . e.”*.
Aytan a 4-log sin y
log cosa – a coty
Aſ a - y g”
37. I_*... 56. —H· .
a; a + 'y a – aſy loga,
2
v (a+b+) y – bº
a: (y – a (a + ba)
|
logio e
2a2
cosec a' (cº-i-aa; + e / absin bº.
1
250 DIFFERENTIAJ, CALCULUS,
m (1+3*) tan-" a log tan-la:--a: .* 3
(1+3*) tan-ºw (Vºcos Væ-3 sin Va.)
6-22 824 + 54.5 - *
80. & ſ -- - - 4 5 —1 | º
42°43’L2+g Vºx-I (524+4+5) sec 1 a W.
85 1. 2 [Deduce from first result of 83 by putting ta:
logº. Tº T' for w, then using exponential values and writing
2% for e”.]
.7t-H2 (or — *-ī-2 (2 – n+2 ( or –
ss. () Prove H,-ºº-ºº-ººrº), and then
a” (!) – 3) #y” (3-4) +3*(8–9) 'differentiate.
CHAPTER V.
PAGE 55.
I (– 1)*a*ml 2 7, 3 b”. 7, 1
º (aac-i- by-Fl & e (a — a)”. Fl tº * (a - bæ)n+1 *
4. (-1)*abºn, 5 (–1)*(be – ad) cº-inſ
º (a + ba)*H T (cº-Ed)n+1
(-1)*a*n! (–1)*(n+3) :
6. tº 0-1). 3Tº Tºyºtá.(a +a)* '
1. 3. 5...(2m – 3 i
8. (–1)”–1 # !-H+.
(a + a) *
3. 8.13.18...(5n–2) (–1)” a_l pa” (n-1)!
9. 5* º #13. 10 (–1) Tſar Iyº
(a + a) *

1 77,7- 727ſ
- — \ – ??? -
11. 2 |cos(a+ #) 3” GOS (** 2 )}.
sº
º 7?,
12. º, | * COS (**) – 10% cos (3a; +n tan-2) º
Ø n
13. º, {1 – 5% cos (24 + m tan-12)}.
eaw ol 20
14. 2 }. +(a” +4b*)” cos (* + m tan-l *) tº
1 e º
15. J. (2” sin ( 2a: + mir + 4” sin ( 4a: + nT \_ 6% sin ( 6ac -- nT &
4 2 2 2
e3% º – 1 º QT
16, T6 | . (10)” cos (, in fan-l #) – (18)” cos (***)
– (34)” cos (ºr + m tanT1 #) e
# IIIs
(ºr •) SO9
& | trauſs b (I+u) uſs ºo: – 9tlºuis 9 (I-Fu) wº Iw w(I–)
= 0 }.00 = a, alou A
© uſs 6 UIS
g F F (U eloq A
(***) SO9) (?- 0) SO9)
Jl. ll.
& 8
‘{brºups b (14-w) us – 9 ºuts 0 (I--w) us) +*
it! Hºw(I–)
& 8
—4–=w otou A ‘9 tº ºuts 0 (I--w) uſs ſº
'6 joo =a oroua ‘94, uſs gu uſs I (I - ºt) t—w(I–) &
“f 100 q= 0 100 p = a alau A
{ | g--wº --- z+wſ! | gº! - zºº
6 tºurs 9 (I--tt) (IIs ºbt+ºus b (I+?!) tus_l intº (I–)
g [..." +a) -- 14.4(? – *| Q&
I T T
|+u(P+*) - Liuſ”- º *] aſ] – 21)
I I I - it w(I-)
‘g Joo p = a, eiau A
*(n + æ) (m) – a p?
& UIIS + 7l) UIS Liu” – Huſ” _ I+1. | 8.
[bºugoſi duº-". I —liu w(I–)
* Fº -º-, &
T I j i (I - u)r aſT-)
30% 6
º-own woº wºulso (I+u) uſs juwg ' ',(I–)
o |* wó) * º it t—ug uſi-)
19 (IðVd
w {-untºuriſm — (a 1–ut, u + æg) uſs ag' d º,
?!, ?ſ, 30
. Jſ & _ ! 3
(# ar) UIS [-wf (#4 zº) ups º
IQZ 'S'ſ ITCHAſ WXMI (THII, Oll, SºHº'ſ AMSAWW
‘a’t 400 = 0 oleuA ‘r-u-(9 Soo-H 9uſs)+
9 (I-Fu) soo – 9 (I-Fu) uſs} 91+ºus iu w(I–) # = *f;
‘9 uſs |(}. 0) uţs g/* =I + a 91auA {(- 6) I +1) UIS 34-wó +
z 8
9 ) } 8+º wº
* + O98O3) UIES —“– = *f;
(. 6 ) I-Lw. {} [-1-wuſ! iwi,(I–)
.* = i+*ſ
I u.
I I I I
* {* Hºo: *—ºv (g – u)(I – u) was “ſi
I I I
g ſº -*). Liu(I - a) +
T T
zºw(I- a , «Puſſ - a) & \ . __ tº
# ****, *, 'ſ--"
e | tºw(p -- a) --- 144(2– a;) } p – 9 = *ſ,
(q – p) (v – p) (0–0)(p − 2)] I waſ[-)
* {(3-1) (I-w) u +ap (I – w) wg +gagpug-F gagp} woog_ºp="fi
'a. Soo 2:28 – ="fi 'g 'a, soo # – a uſs a =*fi
'ga. Soo ga'8; - gº uſs (ZI – tagſ)="fi
'89 (IOVJ
'0 100 = a aroua “{0 (z-w) us (I – wºu + 6 soo 9 (I-u) Us (g-w) wa–
9:Soo guuſs (3 – u)(I – w} 96-wuſs (g-w) t—wſ1–)
"g 100 = a,
9Ieua “{gu up" – 9 up, w) gu Soo 9 Soo 91-guſs i (a – u)r u(I–)
º Hºuſ I + a)
it!
*(q1 + p)=()?-- d arou A ‘(aq Soo () + æquis di) wo
(ºr +.) UIS (I * w) 70 -i- (*. +.) ups arug-H (i. +.) UIS ga:
a 3-w at I-w ll?!,
g—w?
i(g-u)r-u(I –).
''{(I — wy w-H acouz + grap} 209 g—wº '3 *(u--a) 29
'09 CIÉ)VoI
‘ZI
*II
"OT
"Z
'S'ſ) TO OTP/O TWILIAWTAIMH, IICT 69%
A WS WERS TO THE EXAMPLE'S. 253
16. /* = (-1)^ml ſ(n+1)(n+2) 3 (n+1)
º, 4 (a: - 1)n+8 (a: *mº 1)n+2
+++,-,-,--
2 (a – 1)*H 2 (c + irº wº
CHAPTER WI.
PAGE 68.
5
8. *****, e e º 'º
9. (a) tant” ** = tan-1”-tan-is-etc.
q + pa: Q
— or 3
(b) tan-l #. = 3 tan-'a' = etc.
(c) sin-1 1 # = 2 tanT1a; - etc.
— or-1
(d) cos-l : + º =2 cot"a:= T – 2 tanºa: = etc.
CHAPTER VII.
PAGE 89.
8. "0027 of an inch.
CHAPTER VIII.
PAGE 95.
Tangents.
1. (1) Xa; + Yy = c”. (2) Yy = 2a (X+a;).
X Y . . 40
(3) ###-2. (4) Y-y=sinh (x-3).
(5) X (2ay + y”) + Y(a.” +2a:y)=3a*.
(6) Y-y=cota (X-a;). -
(7) X (a”-ay) + Y(y”— aw)=awy.
(8) X {2a: (a^+y”) — a”w} + Y{2y (~24-y”)+ a”y}=a^(wº-y?).
Normals.
_Y X- a Y-y_
Ty (2) 2, t ===0. etc.
2.
4. D/FFERENTIAL CALOULUS,
2.
1
Tangents are Y= + ºx - : e
Normals are Y = + 8 # x+. CL.
(a) |. at points of intersection with a v-H hy=0.
Perpendicular at points of intersection with ha, + by =0.
3 /zº
Parallel at (- a **) e
S/2 2
Perpendicular where a = 0.
4a *#)
(8)
|
Parallel at (
(y) 3 T3
Perpendicular at (0, 0), (2a, 0).
Q} !/ ...:
(r - -- =1.
(a) Tangent, a cos {} + b Sin ()
Normal, aa sec 0 – by cosec 0 = a” – U%.
TT - -- ~~ ..., " . 9 a...,"
Tangent, a sin 3 - 9 cosa-ad sm 3.
(8)
{} . 9 {} . ()
* (U - * — — Nº) - 2 § T. s.
Normal, a cos; +y sing a0 COs 3+2a sing
. A + B . A + B a ..., . A – B
(y) Tangent, a sin -g, 6-y COS sh-0= (A + B) sin 2B 6.
Y
A + B . A + B > A — B
Normal, a cost ºn 94 y sin gº 8-(4–B) cos - 2p* *
For an ellipse, r* = a” cos” 0 + b” sin”0.
For a rectangular hyperbola, r*= a” coš 29.
1 1 1. - 1. , i.e. they must be confocal.
a, b aſ b
The axes are tangents at the origin. Also at the point (2ła, 2ña)
the tangents to the parabolas make angles tan-123, tan-12-3
respectively with the tangent to the Folium.
PAGE 96.
(a) awa- + by, (8) a =0 and y=0, ()) aw– #y VDP-a”.
PAGE 101.
cº/Vaſ-Fºſ. -- 8. al’ea, =; &/away.
m= – 2; n = 1.
gg Z STT TVVVXT THI OI SATTA SAVV
g1 i gº
T-it , e a-sfio-aſia II+ iro-gº
6, %"
'9&I GIOVd
p = = fig + a : pa – ag = ſi
0=z + lizia g/º gi = ſi + a Iz 'g=F = fia; + a : 0= ſig – a
“I=F fi= a : fi - a : 0= ſi '6I 0= I + fi – a 0= fi – a : 0=a.
“I - = ſi – a : I = fi + a : 0= ſig + a LI (ir)r-i o=a
“a = ſi : I F =a gli p = a bI “0= a
“p =F = a 'ZI ſi = a : p = ſi : p = a II p =E = a:
'0= ſi '6 0= fi + a : 0= ſi 0= a 8 0= p + fi + a
“pa - a 9 “O = a “g . 0= fi
0= fi + a: “g 0 = ſi + a: “º, - i-fi+a
ſpa,
“gZI HÉ)Vd
“I =F = a F ſi “g z=a - ſi I=a - fi o = a – ſi 0= a + ſi
“I = a – ſi :0=a – fi : 0= a + fi
'8II I9VoI
'0=I + fig + a F : 0= I + fif + ag : 0= I + ſig + ag
'0= 8 + vg + ſi ; 0= 3 + az + fi :0= I + a + fi :a = ſi
“g/ a F =I+ ſi :0= a + ſi
(I – az) F = ſi : a = fi z ag= ſi ag= fi : a = ſi
'AII I9Vdſ
'XI IGII.CIVII)
. f . , 6
i; og E
'salva-,ſiſ fi-Teulaouqns
tº- ſi º |figq= quo3ue)gns gºleſi = Ieutou gI
6, 9
º - ma “9 ºre-v
'OII IE)Vdi
6%
“0%,
“8TI
9TI
“gTI
“OTI
;
“6TI
“3T
256 JDIFFERENTIAL CA LOUI, U.S.
PAGE 129.
1. 6 = 0. 2. r sin 6 = a.
3. nr sin ( - #) = a sec kir, where k is any integer.
4. r sin 6=a. 5. 7 COS 6 = 2a.
Tr. - (l,
6. 0=;, 7° SII1 0=3.
7. 1 sin ( 6 — #) -: º , where k is any integer.
8. m6 = kir, where k is any integer.
PAGE. 130.
1. a = ſ/. 2. w-y-. 3. w-v=#.
4. a = + 1, y=a. 5. a = a. 6. a = 1, a = 2, y=0.
7. a = 2, y=1, y=a+1. 8. J =0, y=2a.
9. 24-y- + Va”-W2. 10. a = + a.
11. y – a + a = 0, y + æ – a = 0. 12. a = y +2.
13. a = + 2, y – a = 0. 14. -º-ºw/º.
b y++ ſa= --"—
2 Va’ ^ 2 Ma'
16. 3a;+4y=0, y - 2a: =0, y - 2a:--3 =0.
17. 3a;+4y =0, y - 2a: +1=0, y – 2a: 4-2=0.
18. y = + æ, y - 2a: +1=0, y – 23:4-2–0.
15. a = 0, y–w Wa-
19. 1 sin |0. +1) i. - | -: . cosec (2k+1) 5. where k is any integer.
20. r sin 6=2a, r- a. 21. r. sin (1+d)+;=0, 7° E (1.
(l, º
22. T COS 6 = a. 23. g=+cos 0–sin 0.
&
CHAPTER X.
PAGE 141.
1. p = a, p= a cos p; p =3a Secº p sin p: p- a sec p.
2. p=(1+9°)}|64; p=y”|c; p=a sec. 3. p = d.
A VS WERS TO THE EXAMPLE'S.
Z
5
7
11.
PAGE 150.
p= 2, #|ał ; p = a12; p = a”/(m + 1) r".
p=a (0°4-1)*10*; p=a (0°41)*(634-2); p-a/2.
PAGE 158.
(a) p-5 V5/6. (b) p-5 V5/22.
(c) p- - V212 and p-5 V5/18.
(a) p-15a Jäſia and p--" y 2.
a;=7 and a = 1.
CHAPTER, XI.
PAGE 165.
256/3+27a:4=0. 3. a+/a2+b+/y?=cºla”.
1,42 w” 5. (i) 4a3+27ay”=0;
! #30; - 57. (ii) y”=4h (a + h – a.).
y”--4a (a – 2a)=0. 7. Two straight lines.
A parabola touching the axes.
PAGE 168.
27ay”=4 (a - 2a)”. 2. (aa)#4-(y) = (a+-vº.
(a) w”--4ay=0, (8) 4a:*4-27ay”=0, ()) (p − 1) P-la:P+ppayP-1 = 0.
$) g)
2 *4 sº
(n) gº-ſº-dº; (a), (3), (y), etc. being special cases, their
answers may be at once tested by this result.
—“– –“ – ºl,
(y) rº-" — anº-" cos m–n 0; and the results of (a) and (8) may be
verified by this result.
(a) rº–a” cos”0+b” sin”6; (3) r cos 0+a sin”6=0;
(y) a + a = 0; (6) r}=(2a)* cos;
(e) The auxiliary circle; (3) rº–a” cos 26;
3–43 cosé0. iii-xià 70,
(m) rā= aft cos 3 * * (9) ºr = a +1 cos n+1 9.
Similar loci to the results of 6 but of twice the linear dimensions,
E. D. C. 17
258 J)IFFERENTIAJ, OA LOUI, U.S.
8,
10.
ll.
12.
(a) a parabola; (8) a point;
(y) a comic with focus at the pole and of which the given circle
is the auxiliary circle;
(6) a circle through the pole; (e) a circle;
(º) a rectangular hyperbola; (m) an equiangular spiral;
72 %,
#–nions, “ .
(0) ri-n-a-roos H,
&- - - -- 3.
. (a) Wºw!y-We; (8) w8+y}=c&;
+-7
(y) * amy"— Cºm-Fr.
2nt 2n 2n
+
(y) º ... a 2m y2n = cºn-H2m.
1 1. 1. —“– —“– —“—
m) -- m)2(m+n)
(y) ".º: a myº-cºrt".
_^p_ 7??p 7mp
a mºrp-Hymºrp=cm-Ho. 15. A circle.
14.
17.
CHAPTER XII.
PAGE 182.
A parabola with focus at the origin.
A comic with focus at the origin. 3. A circle.
2 \ } * /tºvº -
r}= K*Y* S60 #9. 5. rºtli- *Y. see—tº 0.
2a. 3 0. m+ 1
ºn,
k2\*H* mºmmºn # , ºf k2 \n-l
amy” - (...) (m. Inyº º 7. azº-l-H g” 1 = (#) *
_???, 77. 2007,
(aa)ºl--(by)". T-kº-l.
CHAPTER XIII.
PAGE 196.
Area = 2ab. 15. a”/27. 16. abel3 V3.
4a3/27. 18. pp.4%a Prº/(p + q)P+4. 20. 3 W3/8.
A MS WERS TO THE EXAMPLE'S. 259
21. If a > b Max.-a, Min. = b
If a - b Max. =b. ... e
24. If a and b are the sides the maximum area =} (a + b)”.
25 A Max. when chords coincide with Transverse Axis and Lat. Rect.
(A Min. When chords are equally inclined to Transverse Axis.
PAGE 204.
Max. value=34, Min. = 33.
8. a = -2, -1, 1, 2 give Max. and Min, alternately.
At a = 1, y = Max, ; a =3, y = Min.
At a = 2 and a = 4 there are points of contrary flexure.
10. At a = 2, y = Min. Atre+, y=Max.
PAGE 206.
1. * l
1. (a) Max, for a = 2–-º; Min. for a = 2++.
V3 w/3
(b) a = 1, 2, 3 give respectively a Minimum, a Max., and a Min.
(c) If a > b a' = a gives a Min. ; a = * gives a Max.
If a-v rea gives a Max.; a = * gives a Min.
(d) If a>b, the positive value of y is a Max. when a = (a+4b)/5.
The negative value is a minimum. Also when a = a, y=0
and there is a maximum ordinate for the portion of the curve
beneath the ac-axis, and a minimum ordinate for the portion
above the axis.
If a ~ b the point (a, 0) is an isolated point upon the curve.
2. (a) The greatest and least values are respectively + V a2+ b”.
Max. =q, If a «b ſº
Min
=b.
(b) If a >b | Min. = a.
(c) *=";” *; W(a-b)2+47:3.
C2 a
(d) Max,-i, when tan 6 = + w/ e
(e) Min. = a + b.
260 DIFFERENTIAL CALCULUS,
3.
4.
5.
7.
14.
20.
- -- 3 -
(f) 6–tan-1 §, 5. T – tant” sº, - ***. 37t-
3 & tº dº
2T – tanTl T2 ... give maxima and minima alternately.
(a) was 1 gives a minimum, w=(+ V17–1)/8 gives a maximum and
w-(– VIT-1)/8 a minimum.
(b) wa0 gives a minimum, w= +V3 give maxima.
T . tº 5T . * tº
(c) a = mir-- a give maxima, w=ur +-Iſ give minima.
(d) a = e gives a maximum.
kir
m + 1
and minima alternately, beginning with k = 1; omitting when
m is even those solutions for which k is zero or a multiple of
n+1.
1. 1 £)
(a) The roots of { a Tº b — -º- ) = h^.
7'-
(e) The solutions of ac- , where k is any integer, give maxima
(b) Min. value=20 Val).
(c) Max, and Min, values respectively = + 2 sin 3.
(d) Max. and Min. values respectively = 1 + cosa.
1/"Jp. 6. “P , -47 .
p-H q' p + q
A Max. When the segment is a semicircle.
A Min. when the radius is infinite. -
If height of cone be h and semivertical angle== a, Max. Volume
of Cylinder = # Th9 tan” a.
Half the triangle formed by the chord and the tangents at its
extremities, or three-fourths of the area of the segment.
CHAPTER XIV.
PAGE 212.
(l, 3 1.
logba. 2. 5 3. 2. ' 4. 2. ' 5. 1.
2 1 2 1
3 * 7. 4. 8 3 * 9. 5 10. 4
#
G
~ *
'9% -apºue TU311 tº JT&EI
º/*-arm
Q 8
10
qz/* a = ſi UTIA UOſloes.Iajuſ Jo Slupod alſ, ſº put, UISITO alſº ºw
lº/ 3
“4-g—uſe (3-tu)(I - ut) u%), – 1-ºut'), E 6) pub
“gº
*3%
“ - F-ºw (g – ul)(3 – u)(I-ut) u"ou--a-wºw (I-10) uſow – waſ Ed 8.19UIA
ſ 6 - 2 \, - (6...o.
(***) SO9) o (#42) UIIS diſ
e ſº + æ) - 14.4(3+ * &
I'll w
T T (I–)
i. #; ‘8T 'a(I- a) g
º ‘OT º (AI) : I (III) ; I – (II) : (i)
º : () to (2) #2 (i)
'SQLTJIN.W.XGE SOOGINWTTGIOSITN
ºgº (9) ‘sº (q) #2 (p) “fa ‘go '86 '#2
. GI gº ... 8 - * A – 8 6 – 2 , 6 — I
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262
DIFFERENTIAL CALCULUS,
57.
58.
59.
61.
63.
66.
67.
68.
69.
72.
75.
(i) a + y =0; (ii) a = 0, a = 2a.
a =2y – 14a, a = 3y +13a, a -y-a, a - y=2a.
1. --
0, 1, ~ 3. 60. When y= + v Mn.
Maximum area = 4r” sin a cosła, where r is the radius of the circle
and 2a the given angle.
The Centroid is at the centre. 65. d.
If m be even & =-“t, and a;=(2\+ 1);, where k and X are in-
n+1 2
tegers. If the several angles be arranged in order of magnitude
we have Max. and Min. alternately, beginning with a = 0.
If m be odd the solutions which give Max. and Min. are those of
*=#1 , and we have Max, and Min. alternately, beginning
with ac-0.
a;= 0 gives a maximum. Then the series of angles º in
+ x/7 – 1
order of magnitude) defined by sin a = 0 and by cos ac- 6
give alternate maxima and minima.
T
Alternate Max. and Min. are given by a = k1 + 4
k = 0 gives
8, Iſla X.
If the breadth be 2a the minimum length of crease is º and
tº e 8a?
IIlllllllll-III] area, E — ---...- .
3 V3
a > b Max. if a. -**,
a ~ b Max. if a = a,
a = b gives a point of inflexion.
y” (a +16a)*-i-4 {69°– (2a–w)*}{y”–3a (2a – a.)}=0.
CAMIBRIDGE: PRINTED BY C. J. CLAY, M.A. & SONS, AT THE UNIVERSITY PRESS.
ſiliji
7689


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تهمين******به من مجهة" : " ،* * * يفة "نه ي ة " ، أن مهمة هامة أمام القوة هذه مهمة لانه تم انها لم تنضم اليه
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* * * * * * * * * * * * *.ي م ي ي م . م . و اليمينية بما يمنع
بنيته وفعاليات عند نهاية مهلة تنتهي عند هذا هو ممتنوعة من مه بين فينة ومع
8و " ن . ن ) ؟ف ي ه ع ي ه . . . . . . . . . و
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كلمة له خلال دورات مياه الموجه في نهاية هو نوه بوزوي قيتولي مذهجن هوة ، و
-- - ل م ع ل ي م ي ه يومية في. -تحذيرات مما يجمعياريبيروت
له : يا
مما هي هي ولا :