f(a), if c be positive,
< /(a), if c be negative.
If .-. c be positive .c = a, makes u = b a, minimum,
c be negative x = a, u = b a. maximum.
CHAPTER VIII.
FUNCTIONS OF TWO OR MORK V/VRIAIJLES — IJMl'LK IT
FUNCTIONS.
101. As yet we liave only treated of functions of a single
variable; we next proceed to the case in which ic=f{.i"y),
where x and y are independent of each other, and the value
of u corresponding to new values a: + h, and y + k, of .v and
y, is required.
Now when u is a function of a; and y, u may vary on three
suppositions; 1st, x may vary, and y remain constant; 2nd,
y may vary, and x remain constant ; and 3rd, x and y may
both vary together.
Thus suppose u = xy'^, and let x become x + h, and y re-
main constant ; therefore if u be the value of tt,
u - {x -\- h) y" = xy"- + y^h.
Next let y become y + k, and x be constant, and let w, be
the value of u ;
.-. z«, = x(y + kY = ocy^ + 2xyk + xk^.
Again, in the equation u = wy' write x -{-h for a?, and y + k
for y, and let Uo be the value of w, or ii,., =f{x + k, y + k) ;
.-. Ma = (^ + ^) iy + ^y = ^y^ + y^f^ + '■Ixyk + 2ykh + xk'^ + k'^h,
the same result as would have been obtained had we put
y A- k for y in u, or x + k for x in u^ .
102. Next considering the question in a general point
of view.
Let u=f{x, y), then if y remain constant while a- be-
comes X + h, we have, by Taylor's Theorem,
drt d'u h^ d^u /i"
•' ^ -^^ dx dx' 1 . 2 rf.x" 2 . 3
or, if X remain constant while y becoujcs y + k,
du d'u fc'^ d'u k""
f (x, y + k) = u + -— k + — + -— : . - — + &.c.
•^ ^ ^ ^ dy dy' I .2 dy" 2 . 3
Suppose now that x and y both vary ; or x become x + A,
and y become y + k; it is not possible to make both these
assumptions at once: but if we use either of the two scries,
106 FUNCTIONS OF TWO VARIABLES.
for f{x + /t, y) or /(.r, y + k), and in the former put y + Ac
for y, or in the latter x -\- h for .t', we shall in either case
have f(w + h,y + /c), and its true developement.
Assuming the first expansion,
fix + h.y) = u + - . h + , + — -. + &c.
du d^u . .
But u = f'(osv). and therefore — , , are also func-
• dx dx^
. , du d'u
tions of X and y, if therefore y become y + fc ; m, — - , -— ,
d X d X
&c. will become functions of y + k, and may be expanded by
Taylor's Theorem, x being considered constant.
Let therefore y become y + k ;
du d^u k- dhi k^
.-. u becomes u + —- .k + -— ., -^ + j— , . — — + &c (a),
dy dy~ I .2 dy' 2 .3
du d'u . du
and to obtain the values of -— , - — ; , &c. we must write ^— ,
dx dx- dx
, &c. for u in the series (a) ;
dx'^
d (^\ d' {—\
du du \dxj \dxj k^
.'. — becomes — - + ; k + — —-t — . + Stc.
dx dx dy dy" I . 2
d (~
d'u d^u ' [dx'j ,
-—o -r^ + — ■; •^' + &^"-
dx dx dy
d (— *^
tfw d^'u ' \dxV ,
dar dx dy
d^u ^
But it has been agreed to write , — tor
° dy . dx dy
which expresses that the function has been differentiated twice,
Jst considering .r, and then y as variable ;
/•(I?). . d'u /"j^. ,,
and IS written . , „ , and ; is written
dy dydv dy
.'du
d
du\
d7vl
FUNCTIONS OF TWO VARIABLES. 107
^ — , denotino; the differential coefficient when the func-
rf^/" . dx'" ^
tion has been differentiated m times with regard to .r, and
n times with regard to y.
Making these substitutions, and multiplying the expan-
„ du , , ^ d~u , A"^ „ , „ I
sion of — by /i, that of — ;, by , &c. we shall have
dx ^ dx"" ^ \ .2
du d^u le d'u Jc'
•^ ^ ' ^ ^ dy dy' 1.2 dy' 2.3
du d^tc , , d;^u hk- „
+ -— h+ hk + -—7. — - . + &c.
dx dy . dx dy . dx 1.2
dPu li~ d^u h^k
+ -7-^ + , — z-^ . — + &c.
dx^ 1.2 dy . dx- 1.2
d\i h'
+ -7-3 + &c.
dx 2 . 3
+ &C + &c.
103. But this developement was obtained, by first sup-
posing X, and then y to vary ; but manifestly we should
have had an equal result, had y first become y + k, and then
X become x + h. On this supposition we have
d?( rp7f, k'~ d'u kf
f{x, y + k) = u + -~k + -— — + -— + &c.;
•^ dy dy' 1.2 dy^ 2 . 3
put X + h for Xy
du dru K^ d^u /r
.-. u becomes ^« + -— A + — + - + &c.,
dx dx^ I .2 dx' 2.3
du du d~u d^u Ir
— 1- h + . 1- &c.,
dy dy dxdy dx^dy 1.2
d^u c^u d^u h
— ^ + 5 - + &c.,
dy^ dy dxdy l
+ &c.;
whence by substitution the total developement becomes
du d^u h' d'u h'
fix + h, f/ + k) = u + -—h + —-r, + —- + &c.
•^ • ^ dx da^ 1.2 dx'2.3
du d^u df 11, h^k
dy dxdy axrdy l . 2
108 DIFFEKENTIATION OF
d'u k' d?u kVi
+ -T-z + -; — 7-, — + &c.
dy^ 1 .2 dxdy- l .2
d^u k^
+ 3— + &c.
• dy^ 2.3
+ &c.
Cor. 1. Since the series are equal, the coefficients of the
same powers of h and k ought to be equal ;
d~ u d'u
dydx dxdy
d^u d^u
and
dy'dx dxdy''
d\i d'u
dydx^ dx'^dy*
&c. = &c.
d"'+".w d"'+"ic
dy"'dx'' dx"dy'"
Whence it appears that the order of differentiation is indif-
ferent; or that the differential coefficient of u differentiated
m times with respect to x, and n times with respect to y,
is equal to the differential coefficient when u has been first
differentiated n times with regard to y, and then m times with
regard to w.
, . d^u d~u . . du ^
Cor. 2. Again, '.• - — — = -1—7-'^ •'• writing -— tor u,
dydx dxdy ax
dydx
d'u
or —
dydx^
and
dydx dxdy
d^ .u d\ u
dydxdy dvdy
d X' d X
the consideration of x alone being the inilependent variable
, ^ . , . du d u d u , , , . , ,
10k hincc — , - , — - , &c. have Invn obtained by
dx dx' dx
FUNCTIONS OF TM'O VARIAULKS. 109
such differential coefficients have been called partial diff'er-
ential coefficients, and tor the same reason -— , , &c. are
ay d y'^
also called partial differential coefficients, and these partial
differential coefficients are frequently included within brackets,
thus ( — I is the partial differential coefficient with respect
to -r, and [ — ) is the partial differential coefficient with
dyl
1 (du\ , , (du\
respect to ?/, and I— -Id.c, and \-—\dy, are the partial
differentials of m, with regard to cc and y respectively.
105. The term — h + —A:, which involves only the
doe dy •'
first powers of h and k is called the total differential of w,
and putting dx for /«, and dy for ^•, is thus written ;
or the total differential of u =i f{xy) is the sum of the partial
differentials.
106. Fi'oni the first differential of m, we may form by
differentiation the successive differentials d^w, d^it ; &c.
idu\ , idu\ ,
And differentiating, considering ( — j and ( — 1 as func-
\d,vl \dy/
tions of X and ?/, and dx and dy constant, we have, by
(dn\ jdu\ . , r,
writing successively, |— I and ( t~ 'or u in (p),
, ldii\ /d^zi\ cf?/
Then substituting these values, since
-, dru , ^ d"M , , r^ii , „
a" 7/ = — :; . d.v + 2 . — - . dy . d.v + — . dy.
rf.t" da; . dy ' dy~
110 DIFFEREiNTIATION OF
Again, to find d^ii, substituting as before
, / d^u \ d^u , d^u
d . = — — . dx -\ . dVj
\dxdyl dx'dy dydyd.v
, /cfwN d^u , (d^u\ ,
\darj dardv
+ 3
dw^dy
d^u , „ . [d^
''y'-d-^*{j^)dy:
dy'^dx \dy
107. The law of continuity is almost obvious ; for the
numerical coefficients appear to be those of the terms of the
expansion of the binomial (A + k)" : but to prove that they
do follow this law, assume that
d"M d''7i
- — dx" + n . —-
dx"" dx''-^dy
d"u = -j-^ dx"" + n . 3-737-— . dx"~^dy
n - 1 d^u , „ , .,
+ n. ~-~dx"~-dy- + &c.
2 dx"~^dy^
Then, differentiating the successive terms by means of
du du ,
du = — dx + -— . dy,
dx dy
, (d"u\ d" + ^u , d"-^'n ,
''■[d:r'}'d^*<-''-''*d^r'''' ^'''
/ d"u \ d"^^u d"-*-Ui
\dx"~^dyl dx"dy dx"~^dy^
, / d"u \ d" + ^u , d"+'M , , ^
''■ [d^df] -^d^df-^^'^di^'w " ^''-
&c. = &c.
Then, multiply (l) by dx", (2) by n.dx"~^dy. (3) by
"n, — \\
dx" 'dy-, and addipg
xyz d^u
dxdz (a- - i^)^ dzdx '
dru 2x^z d^u
d^z " (^^^y ^ dzdx '
rf^w 4>xz dhi dhi
dxdydz {a^ - z^f dzdydx dydxdz
(8) Let u = ■ , , sjiew that
\/(a - xf + {b- y') + (c - ^)-
d~u d-u d'u
Jx'^ dy'^l^^^^'
Here — = {a - xf + (/> - yf + (r - zf ; (l) ;
du
.'. ,— = u' (a - x) :
dx
d'u du
.-. -— ; = 37i^ — (a - x) - u^ = Siv' (a - x)' - u\
dx~ dx
EXAMPLES. 115
and from the symmetric form of (l),
d u
-—= 3u'{h -yf -u\
dy'
d?u
(Pit dru d^u ^ 1 „
••• ~r-. + -r^ + -r-o = ^^^ — - ^^ = o.
d.v~ dy dz~ u~
an equation of great importance in physical science.
112. The following theorem is important in the integra-
tion of homogenous differential equations.
Let 11 be a homogenous function of w^ y^ z, &c. and let u
be the sum of the exponents in each term, then
du du du
nu = — w + —r-y + -r"^ ^ ^^•
dw dy dz
For tt', y, z, &c. put x + mw, y + my^ z + mz, &c. then
u becomes (1 + myu ;
du d?u m^x~
.'. (1 + mfu = ^f + -— m,v + -— . — — + &c.
^ ' dw dx^ 1 . 2
du d'u ^
+ — my + - — —m-ocy + &c.
dy dydw
du d~u
+ — mz + - — ~ . mrzx + &c.
dz dzdcc
d~u m^if
dy" 1.2
+ &c.,
w- 1
also = 7^ + num ■>- n iim' + &c. ;
whence by equating the same powers of m,
du du du
nu = — X + -r-y + -— ^ + &c.,
dx dy dz
d'u „ d'u ., dHc ,
and n(n - l)u = -z-^oc^ + -ri. V" + ^r9^' + ^^'
dx- dif dz^
d'u d'u d'u
+ 2 xy + 2 - — — zx + 2 - — , zy + &c.
dydx dzdx dzdy
116
EXAMPLES.
(1) Let 7/ = (v + y + z)' ; hero n = S;
du dii du
d,v dy di
3(a? + t/ + zf;
du du du ,
— .V + -J- y + -j-z = 3 {x + y + zy {.v -i- y + z) = 5u.
dz
(2) Let u =
xyz
00 -\- y + z
; here n = 2,
, du du du
and —i- 'V + ~-y + ~- z = 2u.
doc dy dz
(3) Let u = ; here w = 2,
oc-y
du du
and — 0? + -— y = 2u.
doe dy
(4) Let u
,V' + y~
here n = - 2,
du du
and -r-x + -r—y
da: dy
2m.
(5) Let M = ^^ + ^^ here^=-i
,x +y ^
du du
and -— ,T +-—?/= -iw.
dx dy -^
(6) Let u =
ax + bz
here w = 1,
rfw dw du
and -—0?+ — ?/+ -T-% = u.
doc dy dz
(7) Let u
V "^^ ^ ; liere ?^ = 0,
,r + V
and -^ X -\- —- y = 0.
d X d y
(8) Let u = (,r- + ?/)-; here « = 4 ;
d^M , d^u d^u .,
d.r'^ dydx ay
IMPLICIT FUNCTIONS. 117
IMPLICIT FUNCTIONS.
113. When there is an implicit function of y and x, it is
frequently impossible to solve the equation with respect to y,
and obtain y =f{^) ; but by considering f(w, y,) =0 to be
a function of two variables, we may from the preceding expan-
sions for such functions obtain rules easy of application.
Let u =f{x, y) = 0, and let u^ represent u when x be-
comes 07 + A, and therefore y becomes y + k\
(du\ (du\
But •.• M = 0, whatever x and y are; .-. z«i = 0;
'du\ . idu^
~y>
dv
But k = -^h + kc., '.- y=f{x)\
dx
therefore, substituting for /c,
\\dxi \dyi dx\
ldu\ ldu\ dy
\dx) \dy) dx
fdu\ /du\
a theorem by which — may be found from the partial dif-
dx
ferentials
(du\ _ (du\
' UJ """^ id,)-
dy
Ex. y^ - Saxy + x'' = ,• find — .
Let u = y^ - Saxy + x^ ;
... (-)=-3a!,+ 3.-; y=3, -3«.r.
fdu\ ldu\ dy _
\dx} \dy) dx
dy dy ay - x^
- Say + Sx^ + (3«^ - Sax) -— = ; .•.-—= -^ •
^ ^ ^ ^ dx dx f - ax
118 IMPLICIT FUNCTIONS.
— I do; + ( — ) . — dx = 0; putting da; for h ;
dwj \dyl dx
"' ir^""^ {£)"''"'■ «'•"«!'=/(*)•
'du\ . /du^
,dy>
Cor. 2. Hence, since if w = 0, dw = ; .-. if du
d^u = 0; and thus if m = 0, d"w = 0.
'du\ (du\ dy
114. From tht
(du\ ldu\ dy , ^
equation — + _ . ^ = o (l), to
' \dccj \dy) dx ^ ^'
find — - (where d^u means the second total differential of w),
d.v^
d'-y
and thence to deduce — -.
da,"-
du (du\ (du\ dy
Smce —
''\ dy
dx \dxl \dy) dx
d^u d idiiX d { ldu\ dy\
" dx' ^ 'd~v' \J~i'l ^ d^\\dy] Jx] ^~^
T^ • / . ^w ^ - , du „
rut in (1) — tor w, and then — lor u.,
dx dy
d (du\ /d-u\ d^u dy
dx'Kdxl Vdx'j dxdy'dx^
d i idu\ dy\ /du\ d" y dy d (du\
dx\\dyl dx) ^dyl dx'^ dx dx \dy)
'du\ d^ y dy j d'u du dy)
\dyj ' dx' dxydxdy dy^ dx)
.'. substituting in (2), we have
d-u /d^u\ ^ d^u dy idu\ d^ y jd~u\ dy' _
dx'~\dx-) ~' dxdy' dx \dyl' dx^ \dy^l' dx''
and because from equation (1) — may be found in terms
of the partial differential coefficients ( — 1, ( — ), and (-— -o)
* [dxr {dy! \dxV
(d^u\ d~u , . ■ ., , n 1 ^^ y 1 I
and -- g J ^"" 1 — r being similarly found, - ^ may be de-
\dy / dxdy ' dx~
termincd. In tlie same manner .. , and .-. ., and dillc-
dx dx
IMPLICIT FUNCTIONS. I IQ
rential coefficients of higher orders may be found ; hut the in-
vestigations lead to equations of great complexity, and which
in practice are seldom needed.
115. Next, let w = be a function of three variables
,r, y, ^, or let % be an implicit function of (.v, y) ; and let
^ + m be the value of z when the independent variables, a;
and ?/, become respectively .v + h and y + k ;
.-, since w, = =f(v + h, y + k, z + m),
'du\ , /du\ . fdu\
s.dy)
But z + m = (p {x + h, y + /c);
i'dz\ , fdz\ ,
therefore, substituting for m, we have
{ idu\ (du\ dz\ , ( fdu\ (du\ dz\ ,
"=(u) ^ y -4 ny ^ y -s^r ^*"^-
whence, since h and A; are independent, we have
'du\ dz
— 1 . — =0 (1),
\dzl doe
[du\ . idu\ , (du\ ^.„ ^ .
= I —J h+ (—1 k + I —J w + Ah^ + 5A;- + Cm- + &c.
dyl \dzj dy
dz dz
whence dz - — dx + — dy may be found.
dx dy J '
116. The differential coefficients of the superior orders,
can be found by differentiating the equations
(du\ /du\ dz
y Md-J-dr.'" ">'
du\ (du\ dz
ry)*[di}-iy-' ^''-
-.1^ ™, ^ . d^ z d^z , d'^z
117- Thus to obtam -— , - — ~ and — ;.
dx'' dydx dy~
We must consider equations (l) and (2) as functions of
.f, w, z, and that ( ^— I , \-^\ and (— -] are also func-
\dx) \dyl \dz)
tions of the same variables.
120 IMPLICIT FUNCTIONS.
(1") Let equation (l) be differentiated with respect to
w; it must be considered as a function of x and x, and there-
fore from (Art. 114), putting z for y,
ld-u\ _ cT'u dz /d^u\ dz" idu\ d^z
WJ ^^'dsrTrf^'rf^"^ \d7'j 'd^^"*" Uz] ''d?^^'"^^^'
(2") Differentiate (2), considered a function of y and z.
Write y for w in equation (3) ;
td^u\ d^ti dz ld-u\ dz'^ ldu\ d'z
\dy) "dzdy'dy \dzV dy' \dz) ' dy'^
Now either differentiate (l) with respect to y, or (2) with
- . . - „ dz ^ (Pz
respect to x; and smce in the former case — becomes ,
dx dydoj
11 -11 dz d?z , , drz
and that m the latter — becomes , and that
dy dwdy dydx
= -, the results will be identical.
dxdy
Let equation (l) be differentiated with respect to y% and
to do this put ( — ) and ( j— ) for w in equation (2), whence
we have
d?u d'U dz d-u dz
doo.dy d.v.dz dy dz.dy dx
ld'u\ dz dz /dxX d'z
■^ Vrf7^' 'dv'dy'^ [dTzI'dyZd^^^'"^^^'
r, 1 . / . X . . V <^'^ d.-z , d~i
From the equations (3), (4), (5), -— ;;, — :, and
dx"' dy^ dydx
may be found, while from equations (1) and (2), — ^ and —
dx dy
may be found.
118. From this it is obvious, that if u =f(x, y, z), be
an explicit function of x, y and z, where ;? is a function of
X and?/, that the differential coefficients of u may be found.
du {du\ /du\ dz
Jx " ' d^ j "^ [dz] ' d.
dti fdu\ (du\ dz
^ wt* (du\ (dii\ dz
dy Kdy) " Kdzl dy ^~^'
IMPLICIT FUNCTIONS. 121
dru _ /tf M\ ^ d-u dz^ ld'u\ dz" ' ldAi\ ^
J?~ tej '^^d^d^'dw'^ U2V 'dx''^ [dz! 'dx' "^ ^'
d^u d^u d^u dz /d'u\ dz- (duV d^z
1 2 + I I + j — I . (4)
dif dy' dzdy' dy \dz"J ' dy" \dz) dy^
dru dru d'u dz d'u dz
dwdy dxdy dxdz dy dydz dx
(d-u\ dz dz dx d^z , .
\dz I dx dy dz dydx
,. dz dz d'z d'z d'z , . ,
From which -, — , — -, — ^, , — — may be found.
dx dy dx^ dy~ dydx
By a similar process the differentials of the third and
higher orders mav be obtained.
ELIMINATION BY MEANS OF DIFFERENTIATION.
119. We have seen that if a constant quantity be con-
nected with the function by the signs ±, it disappears from the
differential coefficients. Should it however be multiplied into
the function or any term of the function, it will still appear in
the value of the differential coefficient.
Thus if ?* = be a function of x and y, involving a con-
stant a, both u = and du = will contain a, but between
these two equations it may be eliminated, and an equation will
arise independent of a, which is called a differential equation.
mi 1 V ^y ~y
ihus, let y = aii'^; ,-. — = ^ax^ = — ;
dx X
an equation from which a has disappeared.
Irrational and transcendental quantities may also be elimi-
nated by differentiation.
Thus, let y = (a- + af)" ;
dy m o ..7-1 2ma?(a^ + a^)"" Zmxy
2x. — .{a" ->f x^) -
dx ' n' n{a- + x") n (a^ + x^)
If there be two constants as a and b involved in the
c(puition y = f{x), then to eliminate them, the equations w = 0,
du = 0, and rf-w = must be combined.
122 ELIMINATION OF FUNCTIONS
Ex. 1. L.ct 71 = y - ax~ - ha; = 0, ov y = ax~ + b.v ;
^y c I. ^~y r. . dy dry
dx dx- dx dx^
x^ d-y dy d^y
d^y 2 dy 2y
dx'' X dx x^
Ex. 2. Let y = a . cosmx + h . sin m.r; eliminate a
and b.
dy ,
-— = - ma sin mx + mb cos mx,
dx
d"y
---„ = - nr a cos mx — m^b sin mx
dor
= - w- {a cos mx + b sin mx\ = - m^y ;
d'y
.-. ~~ + m''y = 0.
dx^
Ex. 3. Let y = ae^'^sin {Sx + 6) ; eliminate a and 6.
-I- = 2ae^' sm (3a? + 6) + Sae^^ cos {3x + ft)
= 2^/ + 3y cot {3x + ft),
— — - = 4ae^^ sm (3ct + ft) + 6ae^' cos (3 a; + 6)
dx^
+ Gae^'^cos (3a? + 6) - 9ae-' sin (3a? + ft)
= - 5y + 12y cot (3a? + ft)
dy
rfa?
d-y dy
dx"^ dx
120. Again, if ji = f{xyz) = 0, or z = f\xy).
We may by means of the partial differential coefficients
— and --— eliminate two constants from z = f(xy). and by
dy dx •' ^ •" ^
proceeding to the second differential, we have three other cqua-
rf~ z drz d^ z
tions for -— ;; , and — -, and therefore five constants
dx- dydx oy^
BY DIFFERENTIATION. 123
may be eliminated ; and not only constants, but indeterminate
functions.
Ex. 4. Let x=f(ax + by); eliminate the arbitrary
function.
Let (ax + by) = v ; .-. z =f(v),
^ d% dz dv dz »,, . , dv
dx dv dx dv dx
dz dz dz dv ., , . ,
dx '' ' dy dv dy '' ^ ^
or h . — = ahf'{v), and a . —- = ab 'f'M ;
dx dy
dz dz
.'. b a — =0, or bp - aq = 0.
dx dy
As an example. Let z = sin (ax + by) ;
.•. p = a cos (ax + by), q = b cos (ax + by) ;
.-. bp - aq = 0,
and similarly if ^ = (ax + byy, or ^; = log (ax + by), the
differential equation will be verified.
Ex. .5. Let z = (x + yy"(l)(x~ - //) ; eliminate the func-
tion.
p = m.(x ^ «/)'"-' (p (x^ - y-) +2(x + yY'cp' (ar - y^) . ,v...(\),
q = m .(x + y)'""' (p (x~ - y") - 2 (x + y)'" (p' (,%" - y^) . ?/...(2).
Multiply (1) by (y), and (2) by x, and add;
.-. yp + xq = m (x + y)'" \o) + V^'(«)}t^>
ax u,x
-l = (p{a)+ {xf'{a) + y(p'ia) + v//'(a) ( -^ ;
dy ay
dz dz
therefore from (2), -— = f(a); —-= (b («) ;
dx ' dy ^
dx \dyl
,dyj
dx'^ \dy/ dxdy
"" f\^.%^ w,
dydx \dyl dy'^
whence multiplying crossways,
\dxV KdyV \dxdy)
wliich is the e(juation to developable surfaces
EXAMPLES. 125
Ex. 9. y = xe'''' ; eliminate c.
00 dy - ydx =y (log y — log .r).
Ex. 10. Eliminate a and h from y" = a,v + bx'.
Ex. 11. If y = a sin ,2? + 6 sin 2. r, shew that
£^'£^*''=''-
Ex.12. If j^ = - +/(^ + logcr) ; px-q=x^.
Ex.13. If;? = /f^^ J; ^xyp+ {x' ■\-y^)q=0.
Ex. 14. x^ jf- y^ jf z^ = f{ax + by + c).
(y - bz) p - (x - az)q = bx - ay.
Ex. 15. z = ax + by + c; eliminate a, 6, c.
d?z dry d^z d?y
dx^ ' dx^ dx'^ dx^
Ex. 16. If z = '■»/(-) + (pixy), shew that
drz
d? '"^ ' dy' '
''"'•^"=y'-:^
CHAPTER IX.
MAXIMA AND MINIMA OF FUNCTIONS OF TWO VARIABLES.
121. If u = /(.^', y) be an equation between the function
?<, and the two independent variables, x and y, there may
be some particular value of x, and also of y, which will make
the function greater or less than the values which immediately
precede or follow it. It is then a maximum or minimum. We
proceed to find the relation between the differential coefficients,
when this circumstance takes place.
122. Let Wi be the value of w, when x + h and // + k
are written for ,v and y respectively ; and Wg the value of u,
when w — h and y - 1c are substituted for the same quantities.
Also put A for — — , B for , and C for — „. Then
dx dydx dy
du du - ,
u,=^u + -—h + -—k + ^\AJi' + 'iBhk + Ck^\ + &c.
dx dy "^ ^
and u., = u - (—h+ — k] + ^ \Ah' + 2Bhk + CkH - &c.
\dx dy I '^ ^ '
Now since the values of h and k may be assumed so small
that, (as long as the differential coefficients — - and — remain
dx dy
finite) the algebraical sign of u^- u and u.> - u will depend
fdu du \ , . .. , .
upon that of the term — A + — A; , it is manifest, that if
\dx dy J
this term exist, u^- u and U2 ~ w cannot be both positive or
both negative, or there cannot be a minimum or maximum
_, „ . . . dii du
of u. Therefore at a maximum or minimum — h -\ k
dx dy
must = 0. A condition which can only be fulfilled, since h
and k are independent quantities, by making — = 0, and
dx
du
— = 0, Hence at a maximum or minimum,
dy
u^-u = \ {Air + "iBhk + Ck') + &c.
/r ,
= \A + 2Bn + Cir\ + he, if k = n/t.
1.2
FUNCTIONS OF TWO VARIABLES. 127
Therefore the sign of u^ - u, and also of u., -it, will
depend upon that of the coefficient of — , that is, upon
A + 2Bn + CriK Hence, this term must not change its sign
whatever be the value of n ; which it will not do, if it can
be put under the form of the sum of two squares, as
(t + a)" + /3'.
Now A + <2Bn + Cn' = ^{CA^ 2BCn + C'ri'}
= ^\CA -B-+ {B + Cny\
-Llc-B^-.C^i^.n)]
which is of the requisite form, if CA be not less than B- : or
to have a maximum or minimum of a function of two vari-
„ , du , du - „
ables, we must first have — =0 and — = ; and secondly,
dw dy
d~u d-u . . I d'u
— X r not less than - — —
dai^ df \dydx
d^u , rf^M , ,
123. It is obvious that — - and -—7 must have the same
d X d y
algebraical sign ; and also if they be both negative, u is a
maximum, if both 'positive, w is a minimum.
If the second differential coefficient of u become = 0, when
the first does, there will not be a maximum or minimum,
unless the third differential coefficient vanishes, and the fourth
neither vanishes nor changes its sign, whatever be the value
of n.
Ex. ] . Let u = 00^ + y^ - ^axy,
du x"-
— = Sx~ - Say = ; .-. y = — ,
dx • ' ' a
du , ., a?'
— = Sy - Sax = ; .• . y - ax = — - ax = \
dy a
.-. X = 0, and x'^ - or = ; whence x = a ; the other two roots
x'
■ are impossible ; and // = — = a ; or = 0.
d-u d-ti ^ d^u
Also -—-^ = 6x, -r^= 6y, and , -— - = - 3a.
dx~ dy^ dydx
128 EXAMPLES.
If .r = 0, A = 0, C = 0, and B = -3a.
If ,v = ff, A = 6a, C = 6a, D = - "a,
AC = 36a:\ and B'' = 9a~;
.-. -x = a, '.' A is +, gives a minimum, and u - — a^ ; w =
gives neither a maximum nor minimum.
Ex. 2. w = w^y^ {a - .V — y).
— = 3 j?^?/"^ (a - w - y) - os^y' = 0,
dx
— = 2.2? V (a - .V - y) - x^'y = ;
dy
.'. S{a -X -y) = w; 2{a-x-y) = y; .'.2x = 3y;
a
.'. 3a - 3x - 2x =x, or x = -.
2a-
■sy
-
■2T,
= y^
or
y-
a
~~3
•. a ■
- X
_
2/ =
a -
a.
o
_
a
a
dx^ ^ ^ ^^ [2 9 6 4 .9j 9
_ = 2..= ('.-^-s^)-*'<'!' = 2|^-e-2-i-3} = -7.
d^u , o 3 «' '^^ "^ «^
d^ = ''^^^^'^-"--^^-'"'^';''^'^^i^"i^"Ti="r2
/. ^C=-, and 5^' = — ; .-. ^Cis>5^
72 144
ff^ d a a^ . . . .
and w= — X — x-= — is a maxnnum, ••• ^ is -.
8 9 6 432
Ex. 3. u = {x+\).{y +\).{z ^• 1), where a'hyc' = yi ;
.-. X log a + y log ft + ^ log c = log ^.
Now -=(j, + l).|. + . + (.. + .). ^} = o,
_.(. + ,). |.+ , +(,+ .). ^1 = 0.
, ds; log a . dz log ft
and -— = - z-^^— , and , = - - ;
dx logc aj/ logo
KXAMPLKS.
12?)
loea
.'. z+ I- {.V + l).-°-=0,
logc
(z + l)\ogc = {x + \)\oga, and .-. (f + ' = a' +
loo; 6
Also 5? + 1 - (v + 1) . ,-^ = ;
log c
id .-.
= h'-"
log a log: ^
Also •.• 2? + 1 = {.v + 1) . -^ = (v + 1) .— ^
Jogc logc
(a? + l) log a - logc
logc
(x + 1) log a — log b
.'. iv log a + (x + 1) log a - log b + (,v + I) log a - log c = log-i4 ;
.-. 3x log a + 2 log a - log 6c = log J ;
log Abe - 2 log a
3 log a
log Abe + log a log (^6ca)
.*. ,v + 1 = = ,
3 log a 3 log a
log (Abea) . log(Abca)
2/ + 1 = ~~ ^' , and ijf + 1 =
3 log 6
(log Ja6c)^
Slog.
27 log a . log b . log c
Ex. 4. Inscribe the greatest
triangle within a given circle,
R the radius.
a, b, c the sides.
be be
But u = — . sin ^ = — . sin (0 + 0),
2 2 ^ ^
and 6 = 2^. sin0; c = 2i2,sin0;
.-. u = 2R^ sin 9 sin cp sin (cp + 0) = maximum
K
130 EXAMPLES.
.-. — = 2/?2{cos0.sin((^ + 6)+ sm9.cos((j) + 0)} smcp = o,
d u
and — = 2J?^{cos0.sin(W) + 0) + sin0.cos((^ + 0)] sin0 = O;
d(p
.'. sin {
= 0, or = (p; .'.30 = tt, and = 60° = 0,
and .•. A = 60° ; and the triangle is equiangular.
Ex. 5. Inscribe the greatest parallelopipedon within a
given ellipsoid.
Let 2oG, 2y, 2z be the edges,
2a, 2b, 2c the principal diameters of the ellipsoid :
sr x~ y^
.'. u = Swyz, and — + — + ^ = 1 ;
c a- b'^
a' b'J'
.'. %"= c' \1
du dz
.'. ---=8yz + 8ya!— = 0,
dw dx
du dz
— = 8wz + Syx — = 0.
dy dy
^ dz X c^ , dz y c^
But -— = , and — = — . - ;
dx z a dy % b"
X' c^ , y^ c'
.'. z . -- = 0, and z • 75 =
z a z o-
z"" x' J ^ 2/' -^^
.-. - = — , and - = - = .-. - ;
c' a^ c^ b^ a-
a f z' J b
8 abc
and u -- — -.^ .
3\/s
v/5'
4
But if V= volume of ellipsoid; V = -irabc
2V j-
.', u= 7^ or w : F :: 2 : 7r\/3.
7r\/3
KXAMPLES. 131
124. In the following examples, some only of the steps
are indicated.
Ex. 6. If u = x^ + y^ ~ ^awy^., find x and t/, when
u = maximum or minimum.
X = ^ n ■s/s ; y = 4z a v'^8,
and .-r = ; ?/ = ; give minima.
Ex. 7. u = a I sin x + sin y + sin (j? + t/)} •
a? = 60 ; y = bo ; give u = 3a - — a maximum.
Ex. 8. Given the perimeter of a triangle, shew that its
area is greatest when it is equilateral.
Ex. 9. Divide a quantity a into three such parts x^ y, z^
that ?* = a?'"^";^^ may be a maximum; and shew that it is a
maximum and not a minimum.
ma na pa
X = ; y = ; z = ■ .
m + n + p m + n -r p m + n + p
Ex. 10. Given the surface of a rectangular parallelopipe-
don ; find when its content is a maximum.
If X, y, z be the edges of the solid :
Surface = ^xy + 2a;^ + 2jy^ = 2a^,
and u = xyz a maximum,
whence x=y = z=a; and solid is a cube.
Ex, 11. If the content of the rectangular parallelopipe-
don be given, find its form when the surface is a minimum.
It is a cube, as in the preceding question.
Ex. 12. Let u = ax + by + cz a maximum,
and X- +'!f + z^ = I; find a?, y, z ;
.-. u =
ax ■\- by + (
Vi-.^'^-y^
du
' dx~
ex
a = ;
.-. az = ex,
du
d~y-
h ^ = 0;
z
.-. bz = cy ;
.-. {a:
■ + b' + c') z'
' = c^x' + f +
z^)
= c^
c
a
y
b
\/a' + ,
b- + c^
y „2 +t' + c'^
v/«^
+ b' +
k2
c'
132
EXAMPLES.
Ex. 13. Given the radius of a circle described about
a triangle ; find its form when the perimeter is a minimum.
If e and (p be two of the angles, and if r be the radius
of the circle,
M = 2r jsin + sin ^ + sin (9 + (p)} ;
whence = = 60 ; and the triangle is equilateral.
Ex. 14. Given the sum of the three axes of an ellipsoid ;
find them when the volume of the ellipsoid is greatest.
If 2.'F, 2t/, 2^ be the three axes,
3
a maximum,
and 2cr + 2y + 2« = 2a, the given length ; whence
X = y = z =^ a.) and ellipsoid becomes a sphere.
Ex. 15. Find that point within a triangle, from whicli
if lines be drawn to the
angular points, the sum of
their squares shall be a
minimum.
Let ABC be a triangle,
and P a point within it,
a, 6, c, the sides of the
triangle.
Draw PN, AD perpendicular to the base; join AP,
BP, CP.
Let CN = x ; NP = y ; then AD = 6 sin C ; CD = b cos C
Then CP~=a;- + f\ BP'= f+{a-wy=y' + ar+ a"- 2a.v,
AP' = (6 cos C - asy + (b sin C - yf
= foa + x^ + ^f -Ob (x cosC +y sin C) ;
.-. u = 3x^ + 3y^ + a' + b^ - 2ax -2b{xcosC + y sin C) ;
.-. a? = 1 (a + 6 cos C) ; y = 1 6 sin C ;
.-. CP = \/^+7 = i \/2 a' + 26' - c\
The point P is the centre of gravity of the triangle.
Ex. 16. Find a point within a triangle, from which if
perpendiculars be let fall upon the sides, the sum of their
squares shall be a minimum.
ABC the triangle as before, P the point within it, draw
PNy PM, PQ respectively perpendicular to CB, CA, AB.
EXA31PLKS.
133
Let CW = a ; PM = p ;
NP = y: PQ = 9;
Then u = y^ + p^ + q^-
Now if ^ be the perpendicular from a point (a, /3) on
a line y = mx + 6,
1^ — b - m a
(1") U S = p; (i = y; a = a'; 6 = 0; m = tanC;
V -
.-. «=;^ ^^ = ycosB - (a - x)smB;
sec B
.-. u = y- + {ycosC - x sin Cf + \y cos 5 - (a - x) sin 5 j',
a sin ^ . sin S . sin C
whence 2/ = ^ ^ j _ ^os^ B . cos^ C + sin 5 . sin C . cos B . cos C)
abc sin ^
and
a^ + 6' + c^
afecsin B
a' + 6^ + c^
9 =
aftcsin C
Ex. 17- Find a
point P within a
quadrilateral figure ^
ABCD, from which
if lines be drawn to
the angular points,
the sum of their
squares shall be the
least possible.
AB = a; BC = b ;
AD = c; AN =x% NP = y;
.♦. u = x~+y^+y~+ {a - xf + {b sinB - yf + (a - X - bcosBy
+ (c sin A - yf + {x - c cosAf
= 4>y- + 2x~ + 2(a - xf -2b [y sin B + {a - x) cos B\
+ b^ + c^ - 2c (y sin A ■¥ x cos A).
134
EXAMPLES.
whence
w
= i(2^
a + c cos A -
■ b cos B),
V
= i(c
sin J + 6 sin
B).
d'u
S
d'u
d'u
u
is a I
do!'
dxdy
Ah
Ex. 18. Of all triangular pyramids of a given base and
altitude, to find that which has the least surface.
Let a, 6, c be the sides of the base, h the altitude of the
pyramid, 0, <^, \|/ the inclination of the faces to the base.
Then because if p be perpendicular from vertex on the
side a, psmO = h, and area of face = ^ ap = ^ ah cosec 9 ;
.*. u = ^ h (a cosec + b cosec (p + c cosec \|/) (l).
Also since the base of the pyramid may be divided into
three triangles, whose altitudes are respectively
h cot 9, h cot (^, h cot \|/ ; if m^ be its area,
m^ = 1 A (a cot + 6 cot <^ + c cot \//) (2),
from which combined with (1) a minimum, we have 9 = (p = \lx,
or the faces are equally inclined to the base.
Ex. 19. Two points P and Q are given above a plane;
find a point ^ in a plane, such that PR + RQ may be a
minimum.
Let the given plane be that of xy; from P and Q draw
lines perpendicular to it, let the axis of z pass through P, and
the axis of ,v pass through the foot of the perpendicular
from Q.
Then if c = the co-ordinate of P, a and h that of Q,
w and y of R ;
.'. u = PR + QR = \/.v' + f + c' + \/y' + (a^xf^^;
du X a — at
d a; y/w^ + t^ + c'^ \/ y' + (a - xf + b^
du y y
dy y/x'+y^ + c^ \/y^ + {a-xf + h'
0...(2).
From (2), y = 0, and therefore the point R is in the
axis of .V.
,v a - X
From (1), _ = ~(\Tf '-> "'' ^^^ cosines of the angles which
EXAMPLES. 135
PR and QR make with the axis of w are equal,
X a — CD ac
also •.• / ., = /. .o =^T^^ ^ =
\/a>' + d" s/{a -wy + W h + c
125. When u =f{xy%) is a maximum or minimum, we
du du du .
must put — - = ; — — = ; --— = 0; and the equation of con-
dx dy dz
dition is {AC - B') (AD - E") > (AF - BEy *,
d'u d^u d^u
dx^ ' dydx^ dy^ '
andZ>=— ^; E = -—~-., F =
dz" dzdx dzdy
Ex. 20. u = ax^y^z'^-x^'i^z^-x^y^z'^-x^'ifz^= a maximum.
1 3a 2a
) may be represented by a series of points; for if it be
assumed that the values of x may be taken along the line Ax,
and those of y be drawn perpendicular to the axis, we shall
have, when x = AN, y = f(AN), which may be represented by
some line as NP. Hence Ax is called the axis of - ^
.-. mx + b - niiX + o, ; x = ,
m - nil
mbi - mb mb^- m^b
and y = 1- b = .
m -m^ m - w,
133. Find the equation to a line passing through two
given points.
Let y = mx + b he the equation to the line where m and b
are to be determined.
a and /3, a, and /3, the co-ordinates of the two points;
.-. /3 = ma + 6; /3i=mai + fe;
.-. )3 - Pi = m . (a - a,) ; .. m = .
But •.• y = mx + b, and (i = ma + b;
.'. y - Q = m . (x - a) = * . (« - a).
a - ai
134. To find the angle which two straight lines make
with each other at the point of intersection.
y = mx + h, and y = m^x + b^,
the equations to the two lines.
PQR and P^QR^ the lines.
From A draw An parallel to Pli,
and Am parallel to P'Ji ;
.-. luAm = ^PQP', A
140 THE STRAIGHT LINE.
.-. PQP' = nJx - mAx = tan"' m — tan"' yw,,
and tan PCIP' = .
1 + w? m.
Cor. 1. If the lines be parallel, PQF = ; m - Wj =
and Wi = m ;
.'. V = tnx + h\ , . ,, , ,.
> , are the equations to two parallel lines.
and y = ma; + 6J
Cor. 2. If the lines be perpendicular,
1 m - /«i
tan PQF
1 + wm,
.-. 1 + nimi = 0, and w^ = - — ;
m
therefore, if y = mx + 6 be the equation to a line, y = ^r + 6,
m
is the equation to a line perpendicular to it.
135. Find the equation to a line drawn through a given
point perpendicular to a given line.
y = moo + 6, the equation to tlie given line, a and /3 the
co-ordinates of the given point ;
.*. y = X + 6i is the equation to the perpendicular,
also /3 = a + 6i, since it passes through (a, /^) ;
•'• (y ~ (^) — (<» - a) is the equation required.
136. Find the perpendicular distance of a given point
from a given line.
y = mx + h the equation to the given line, and (/3, a) the
given point ;
••• (y - /^) = - ^ ('^ - ")
is the equation to a perpendicular from a given point upon
the given line.
Then if S be the distance required, and ?/, and .r, the
co-ordinates of the point of intersection of the given line with
the perpendicular,
^ = Vi'V, - ay + (y, - f^y = (.r;, - «) . \/'''^^^ •
THE STKAKaiT LINK. |4|
But mx. + b = 3 ^- ;
m
/ 2 ,\ n L '^(^ + a - mb
'. ;
a
b y X . ,
.'. y = L_
jl"
->
p
TRANSFORMATION OF CO-ORDINATES.
140. In some problems it is necessary to change the
place of the origin, and the position of the axes, these cases
will be separately treated.
(1) Let the origin be changed.
A the origin at first.
B the new origin.
P a point in the curve.
\ the co-ordinates oi B.
JN=.v, BM = .%,,
NP = y, MP = tjy
Then x = w^-ir a, and y = y^ + /3.
Substitute these values for x and y, and the equation is
transformed, and the co-ordinates are measured from B.
(2) Let the axes be changed but still rectangular.
y
m
K
^
^,--'*i
^
M
^
-^
Acc^ Ay, the old axes, Ax^, Ay^^ the new ones.
AN = x, AM = cc,, NP^y, MP = y,, zxA.v^ = 0,
Draw Mm ± to PN, and Mn JL to A.v;
.'. CB = An - Nn = x^ cos Q - y^ sin 0,
y = Nm + Pm = x^ sin 9 +y^ cos 0. For /.mPM = 9.
(3) If the origin and position of the axes be changed.
Let a and /3 be the co-ordinates of the origin, therefore we have
X = a + Xy cos 6 - yx sin 0,
y = (i + x^smd + 2/i cos 0.
141. To transform rectangular co-ordinates into polar,
the origin being the pole.
AP=r, A PAN =6;
.-. X = rcos6; y = r sin 6,
which put for x and y in the equation.
144
TRANSFORMATION OF CO-ORDINATES.
y
But if the point S be the pole, draw
SB ± A X, and xSm ± PN.
AB = a, SP = r,
BS= B, L PSm = Q ;
.'. 00 = AB + 5iV = a + r cos 0,
y = ^^ + Pm = /3 + r sin 0.
Ex. 1. Find the polar equation to the circle round a
point S. co-ordinates a and /3.
.j?"^ •¥ y^ — a^ ;
.-. (a + r cos 0)- + (/3 + r sin Of --= a^ ;
.-. r^ + 2r . (a cos + /3 sin 0) + a" + ^^ - a^ = 0.
Ex. 2. Transform (a?" + ifY = n^ {x^ - y~) into polar co-
ordinates, the origin being the pole.
,v^ + y~ — r^, and x = r cos 9, y = r sin ;
.-. r*=aV.(cos=0 - sin^0) ;
.-. r^= a^ cos 2 0.
Ex. 3. Transform the equation x^ - y^ = a^ into another,
the co-ordinates of which are rectangular, but the axis of y is
inclined at an z 45° to the axis of x.
X = a?, cos — yi sin 6,
y = Xi sin 9 + x^ cos 9,
1
71
v/i
= 27r - 45; .-. cos
sin 9 =
vi'
y
v/2
^, _ ^ ^ C^. ^ yO- - (^> - y.f ^ t-jy, ^ ^ ^.
^ 2 2 '^'
'iJ'i = X
THE PARABOLA.
145
THE PARABOLA.
142. If from a fixed line QDq perpendicular lines, as
QP, be drawn, intersecting lines equal in g
length, but drawn from a fixed point S,
the locus of P is the parabola.
Draw SD±Qq, bisect SD in A,
then the curve passes through A.
UL
Let SA = AD = a, AN = cc, NP = y.
Now QP or DN = SP ; /. DA + AN = \/NF^+ SN^i
.'. a + X = x/'if + {x - af ;
.-. (a + xy or (x - af + 4-ax = y^ + (x - a)^ ;
.'. y^ = ^ax.
CoR. 1. Ux = a; y^ = ^a^\ «/ = 2a, let this value of
y be 1 Z ; .-. / = 4a, and y^=lx\ /, which is the double ordi-
nate through the focus, is called the Latus Rectum.
Cor. 2. The polar Equation. Let SP = r, z ASP = Q.
Then r = DN = 2a + SN = 2a + r cos PSN
= 2a - r cos ;
2a a
•*. r = = .
1 + cos 2 ^
THE ELLIPSE.
143. If from two fixed points S and i? two lines SP
and P^ be drawn and intersect, and SP + PH = a constant
line, the locus of P is the ellipse.
10 L
146 THE ELLIPSE.
Let SP+ PH= 2a. Bisect SH in C, and take CA = CM = a,
the curve passes through J and M. Through C draw BCb
_L to SH. With centre /S and radius = a cut this line in
the points B and 6, the curve will pass through B and h,
since HB and ^6 each = a ; make CB = 6, and let CS :
CA :: e : 1 ; .•. CS = ae, which is called the eccentricity.
LetCiV=.x; NP=y, SP = D; HP = D,,
,'. D' = SN- + NP" ={ae + xf + y\
D^ = Hl^ + NP^ ={ae- .vf + / ;
.-. Z>' + Z),' = 2 (a'e' + cr' + y% and i)- - jDi' = 4.ae.v.
But D + D, = 2a; .-. D - D^ = 2eai;
.'. D = a + ex, and D^ = a - ex;
.-. D^ + Z>/ = 2a^ + 2e^x' = 2 (o.^e^ + x^ + y") ;
.-. y' = a'. (I - e") - x\ (l - e^) = (l - e'O («' - x"").
CS' a' - CS" SB' - CS" b'
But 1 -
a' ar
CoR. 1. If ^ be the origin. Make AN = x^ ;
.-. a?i = a + cr, or x = .rj - a ;
- .1
2 6%
.-. / = -2(2a.ri-,r,^).
Cor. 2. If 6" be the pole, and ASP = 6, and SP = r ;
.-. (^2a-rf= HP' = HN' + NP~ = (2ac - SNf + r sin~(?,
and ^iV = r cos PSH = - r cos ;
.-. 4a^- 4«r + r^ = (2ae + r cos B)^ + r^ s,in'9
= 4-o'~e' + 4aer cos G + r' ;
_ « (1 -e')
1 + f? cos
CoR. 3. If C be the pole, CP = r, and PCM = 0.
Then x = r cos 9, and // = /• sin ;
THK ELLIPSE.
147
a^ b
ab
ab
v/6' cos'0 + a^ sin' \/«2 (1 - e') cos' + d" sin'0
6
\/l-e'cos'0
THE HYPERBOLA.
144. If the diiference between SP and PH he constant,
the locus of P is the hyperbola.
Let the difference be
2 a ; bisect SH in C Take
C^=a=Cil/, and the curve _.,A--^" ^^ / \
passes through J. - - ^
CN = .37]
. Let CS = e . CA = ea, where e > l.
Then HF^ = /fiV^ + NP = {ea + wf + y~ = D,\
SP' = SN- + NP' = (ea - xf + y- = D';
.'. Di' + D- = 2. {a^e- + x^ + y-) ; Z)," - D^ = 4>aea;.
Also Z>, -Z) = 2a; .-. i), + D = 2e.r ;
.-. D^ = a + e.v, and D = ex - a ;
'. 2d- + 2e^r^ = 2 {a~e^ + .v^ + y') ;
.-. f = (e^ - 1) . ,r' - (e' - l)a' = (e^^ - i) (a;^~ _ «2)
b~
= — 2 . (.^^^ - a^) ; making 6' = a^ (g- - 1) ;
^' _ :!" _ _
•'• h' a'~ ^"
Coil. 1. If ^ be the origin, and JN = xy, .-. a; = .r, + a.
.\ X + a = x^ + 2a; ,v^ - a^ = x^ + 2ax^\
i2
and / = — {2 ax -t x^).
a~
Cor. 2. To find the polar equation, S being the pole,
SP = r, z J^P = 0.
l2
148
THE HYPERBOLA.
Then (2 a + rf - HP' = PN' + UN'
= PN'+ {2CS -SNf
= r^sm^9 + (2ae -rcosOf;
.'. 4>d^ + 4!ar + r^ = r^ + 4a^e^ - 4-aer cosO;
_ a{e'-l)
I + e cos 6
Cor. 3. Let C be the pole; CP = r; i ACP = d.
.-. X = r cos 0, and y = r sin ;
^2 '
4-'
(cos' 9 sin' 6]
[ a' b' )
= 1;
ah
b
\/b'cos'e-a'sm^e \/e' . cos^ 9 - 1
145. The asymptotes being the
»^^
the equation to the hyperbola.
The asymptotes are lines, as CO
and Co, drawn through the centre,
making an angle = tan"' with the
axis of the hyperbola.
CN = a;, CM=.T, and OCJ=oCJ=e,
NP = y, MP = y,.
Draw Mn _L to CAN^ and Pm JL to Mn.
Since MP is parallel to Co, and Pm to CN, .: z MFm = 9.
Now X = Cn + wiV = ii?! cos + y, cos 9 = (a\ + y^) cos 9,
y = Mn - Mm = x^ sin 9 - yi sin 9 = (x^ — ?/,) sin 9 ;
x^ 2/' (vi + yO'
•■• a^- f/^^-^^ '''"'''-— b'
cos'0-^:^LZ^sin^^ = l.
But tan
cos'0
1
co^~9
sin=^ 6> cos=^ 9
. 1 + tan^ 9 =
and
ft=-' + a'
h- + a''' ' h' a'
b^ + o^
THE UYPKRBOLA. I49
i.e. 4.t",y, = a + /r, ci?,?/, =
CoK. If the hyperbola be rectangular, h = a, and .i?i yi= — .
146. The curves whose equations have been just investi-
gated are termed Conic Sections, since they may be supposed
to arise from the intersection of a cone by a plane ; and they
are also called curves of the second degree, since the sum of
the indices of w and y in any one term does not exceed two.
The general equation of the second degree is of the form
Ay^ + Bxy + Cx^ + Dy + Ex + F = 0.
Now if the centre be the origin, the equation to the
curve ought to remain the same when {- x) and {- y) are
put for X and y : consequently the origin of the co-ordinates
of the general equation is not in the centre ; since Dy and Ex
will both change their signs, when { - y) and ( - x) are put
for y and x.
To get rid of these terms, transform the equation to the
centre by putting x + a and ?/ + /3 for x and y, and make the
coefficients of x and y respectively = ; there will be two
equations to determine a and ^ ;
2AE-BD ^ 2CD - BE
^"^"= B^-^JC' ^= B^-4>AC ■
The equation now becomes Jy^ + Bxy + Cx^ + jPj = 0.
Next, to get rid of the term Bxy; let the axes be changed
to others, making an angle 6 with the axis of a?, by putting
X = X cos 9 - y sin 9,
and y = X sm 9 + y cos 9.
Therefore the coefficient of xy becomes
2Asm9cos9 + B (cos'^ 9 - sin^ 9) - 2C sin 9 cos 9=0,
n
or (A - C)sin29 = - B cos 29 ; .-. tan 29 = ,
A-C
an equation which is always possible, since the tangent passes
through all degrees of magnitude from zero to infinity. The
reduced equation finally becomes My^ + Nx^ + Fi = 0, which
may be made to coincide with the equations to the circle,
the ellipse, or the hyperbola, by giving proper values to M,
N, and F,.
150 THE GENERAL EQUATION.
147. To find the relation between M, iV, J, B and C, or
to prove that ^MN = 4 JC - B^.
By putting in the general equation x cos 6 - y sin for x^
and ,r sin B + y cos 9 for ly ; we have
M = Aco^^e - B .smO cos + C . sin^ 0,
iST = J sin^ + fi . sin cos + C. cos^ ;
... M +N = A + C,
M - N = (A- C).cos2e - B .sin 20.
- B
But since tan 20= _,
A - C
A - C - B
cos 2 = — =^ , and sin 2
x/{A-Cf+B'
2iV = J + C-x/(^ -C)'+5^
.-. 4il/iV= (A + Cf - {A- Cf - B'' = 4>AC - B\
Whence, if 4^C> J5^ M and iV have the same sign ;
if ^AC
.-. My^ + {2Mh + R) y + Px + Fa + Rb + Mir + F=0;
-R
and we find (t and b, ■.■ 2MI> + R = 0, or h = — — ,
2il/
(;URVES OF THE SECOND DE(iREE.
4>FM - li
151
and Pa + Rb + Mb"" + F = ; .-. « =
4il/P
and .-. My^ + Px = 0, the equation to the parabola.
If M = 0, we shall have Nx^ + Ry = 0.
CoR. 2. If My- + Nx^ = F be an ellipse, find the axes.
a:'
Let rt, b, be the semi-axes ;
y'
h = '-^
M ., N
but from the given equation "7; ^^ + T^ "^^ = ^ '
F F
... a'^ = — ; b' = — :
N M
»=v/J.-^/|
Ex. 1. Prove that the curve defined by s/y + \/ x = \/o,
is a parabola.
Ex. 2. The equation y- - 2yx + w^ - Hx + l6 = belongs
to a parabola, its axes are inclined at an angle of 45° to the
axis of the parabola, and its equation when transformed to that
axis is {y + \/2y = 2 (w v/2 - 3).
Ex. 3. Find the centre and the axes of the curve
y^ - 2cvy + 3w^ -\- 2y — 4!X = 3.
Ex. 4. Find the centre of « = .
^ X + 1
THE CISSOID.
148. JQB is a semi-
circle. Take AN and BM
equal. Draw the ordinates
NQ, MR. Join AE cut-
ting NQ in P. The locus
of P is the cissoid.
Now
AN'
'nW'
AN=x, NP =
_ AM' _ AM
''MR''
cv' 2 a
AB = 2a.
AM^
AM. MB ~ MB'
r
2a -X
152
EQUATIONS TO CURVES.
Cor. To find the Polar Equation.
AP = r, z PAN =0, X = r cosO, y = r sin 9,
y^ sin^ x r cos
a?^ cos^0 2a -.2? 2a -r cos 9^
.-. 2 a sin^ = r cos (sin^ + cos^ 9) ;
sin . ^ ^ , -
.'. r - 2a . sm = 2a tan . sm 6^.
COS0
THE CONCHOID OF NICOMEDES,
149. The line CP revolves
round a fixed point C, cutting the
line ARN: RP is always of the
same length; then the point P will
trace out the conchoid.
Let CA = a,
AM =
RP = AB=b, MP^y.
MP' _^_ RN^ _ RI^-NP"
'CM' ~ CA' ~ NP"' NP
f \? - x'
(a + xf x'
.-. x'y'={a^xy{}?-x\
Let CP = r, I PCM = 9,
b
r = CP= PR + CR = a+ -.
cos 9
Cor.
THE WITCH
150. AQB is a semi-circle,
and iVP is taken a fourth pro-
portional to AN<> AB, and iVQ,
Then the locus of P is the
Witch.
JiV
AB = 2a,
NP = y\ .-. NQ = \/2ax-x-,
X : 2a :: \/2ax - x' : y ;
EQUATIONS TO CURVES.
153
//
2a \/2a,v — x^
/2 a
\a V —
151. The Logarithmic Curve.
In this curve, the abscissa is the
logarithm of the ordinate, or if a be
the base of the system, the equation
to the curve is « = or* ;
.-. AB = «»= 1,
or the ordinate through the origin is always unity.
It is obvious that as the abscissa increases in arithmetic,
the ordinate increases in geometric progression.
152. The Quadratrix of Dinostratus.
While the ordinate RN of the quadrant
AQB moves uniformly from A to BC^ the
radius revolves from CA to CB^ cutting RN
in P; the locus of P is the curve required.
AN = X,
CB--
= a.
NP^y,
z
QCA--
= 0.
Then e :
TT
2
:: X :
a;
•••
PN
CN a
y
^ = tan
- cF
=
tan
y
= (
a-x)
. tan
ttx
2a
6 =
2a
2a
CoR. When x = a; y = Cb =
153. If RN move as before, and a line as
QPM parallel to AC move uniformly from AC,
the intersection P of RN and QAf will trace
the Quadratrix of Tschirnhausen.
Here AQ : AQB :: x : a ;
o/T/^
i
/
.^^ X ira
AQ = -.~
a 2
TTX
2
. /AQ\ . TTX . ,
a . sin I I = a . sm — is the equation.
154
EQUATIONS TO CURVES.
THE LEMNISCATA
154. If SH be a
straight line bisected in
C, and if SP and HP
revolve round S and
H, and intersect in P,
so that SP>^ HP =CS\
the locus of P is the
Lemniscata.
CN=.v: NP = y; CS
SP = VPlSn + SN^= Vf + {a + w)%
HP = ^PN-'+HN''= \/f- + (a-.vy;
•■• \/y^ + (a + w)- X y/y^ + (a - .vf = c? ;
«';
(/ + a^ + a'^)2
4a-^a?'^= a^
CoR. If CP=:t., and z PCi/ = 0.
Then x = r cos ; ?/ = r sin ; a'^ + ^/^ = r^ ;
.-. r" = 2a~r^ (cos^0 - sin^0) ; .-. r' = 2 a' cos
which is the Polar Equation.
20,
THE CYCLOID.
155. The Cycloid is described by a point in the circum-
ference of a circle, which rolls alone; a horizontal line.
W 6 B
Let BQ.D be the circle, O the centre ; and when its di-
ameter is perpendicular to the horizontal line at y/, let the
point P, which generates the curve, also be at A.
EQUATIONS TO CURVES. 155
Then Ah must = arc /*&, since each point of Ph has
been in contact with each successive point of Ah.
(1°) Let AN = a; BD = 2a,
NP = y, I QOB = 9 ;
.-. X = Ah - Nh = aO - asinQ = a{d - sin 0) ;
.-. y = hm = a ver. sin = a (l - cos B) ;
an algebraic equation cannot be found between .%• and y, but
.,y
since 0! = a ver. sin ' y/^ay - y^
a
dy \/2ay-y^ \/2ay - y^ \/2ay - y^'
a differential equation.
(2') To find the equation from the vertex D.
Let DM = X ; MP = y; Z DOQ = 9.
Join Pb and QB, then these being equal and parallel,
PQ= Bh = AB - Ab = AB- Ph= DQ.
Then y = PM = MQ + PQ = a&mB + a9 = a (9 + sin 9),
.V = DM = a ver. sin 9 = « (1 - cos 9).
Cor. Since x = a ver. sin 9 ; .-. 9 = ver. sin~^- , and
a
a sin 9 = y/^ax - a?-;
.-. y = y/^ax - x'^ + a ver. sin~'-
a
dy a - X
dx \/9,ax — x"
2a - X
a
\/2ax -
- X-
\/2ax -
- x^
y/2ax - x^ 'i'
which is the equation most commonly used.
156. From (2") may be derived a mechanical method
of describing the cycloid ; for the point P is found by draw-
ing MP perpendicular to DB, and equal to the sum of the
ordinate QM and the arc DQ of the circle.
157. If we take MP equal to DQ only, then the locus of
156
EQUATIONS TO CURVES.
P is a curve called the Companion to the Cycloid : its equa-
tions therefore are
y = aO; x = a (l - cosO),
dy a
whence
df yj^-ax
THE TROCHOID.
158. The trochoid is the curve traced in space by a
point B in the circumference of the inner circle BRh^ whilst
the outer circle AQ^ rolls upon a horizontal line.
P a point in the trochoid. Through P draw a horizontal
line MRPm. Take and o the centres of the circk
Draw ORQ. and oP.
Then Pm = RM, and i AOQ = lA.oP.
Let OA^a, AN = xy ^ j^R = 0.
OB = h, NP = y]
Then it is obvious that arc AQ, = AA^ ;
.-. X = AA^ - NAi = aO - b sin 9,
y = NP = oA^ + om = a —h cos 9.
Let e = -
a
X = a{9 - e ?,m9)\ y = a {I - e cos 9).
If e=l, or 6 = a, the trochoid becomes the common
cycloid and their equations coincide.
EQUATIONS TO CURVES.
157
THE EPICYCLOID AND HYPOCYCLOID.
159. If one circle revolve upon another circle and in the
same plane with it, the curve described by any point in the
circumference of the revolving circle is called the Epicycloid ;
but if the revolving circle move within the other circle, the
curve described by the point is called the Hypocycloid.
P the describing point of epicycloid.
158 EQUATIONS TO CURVES.
CA=a; CB^b; zACQ = e;
CN = .v; NP = y; ^PBQ = (p.,
.-. 0) = Cn + Nn = (a + b) cos 6 + b sin PBm.
But PBm = PBQ - (90 - 9) = (p + - 90.
Now AQ = PQ; .■.ae = b(p; .-. = _;
.-.
and y - (a - b) sin - 6 sin ( — — . J ,
or we may derive it from the figure.
For PBm = PBC - nBC = 180 - ^ - (90 - 9)
o
a — b ^
= 90- — .e,
and X = {a - b) cos + 6 sin PBm
= {a - b)cos9 + b cos ( -— . j ,
2/ = (a - 6) sin - 6 cos PB m
fa - b -\
= {a - b) sin9 - b sin I — - — . 9\.
161. If the describing point P be not in the circum-
ference, but within the revolving circle and at a distance 6,
from the centre. The curves described are then respectively
called tlie Epitrochoid and Hypotrochoid : and their equa-
tions arc
.X = (a + b) cos 9 — by cos I . 9\ ,
y = (a + b) sin 9 - />, sill l" ' ■^)\
(0-
EQUATIONS TO CURVES. 159
.(2).
la - h \ ^
,r = (« - 6) cos + />i cos I .6\
y = (a - b) sin - bi sin | — — . 9 j
162. If in the epicycloid a = b,
.r = a (2 cos - cos 2 0),
y = a(2 sin - sin 20) ;
.*. squaring and adding,
a?2 + ?/^= a2(4 - 4 cos 0+ 1);
.•• x" + t/"^ - a^ = 4a^ (l - cos 0).
But a? = a (2 cos - 2 cos" + 1) ;
.•• w — a = 2a cos (1 - cos 0) ;
y = 2 a sin ( 1 - cos 0) ;
.-. {x - af + y" = 4>a^ (l - cos 0f.
But 16 a' (1 - cos 0f = Ot;^ + / - a'f ;
.-. 4a- {y^ + (.V -ay\ = (x' + y' - dy ;
the equation to rectangular co-ordinates.
Cor. If X - a = r cos a^r^ = (r' + 2ar cos 0)^ ;
.-. r = 2a (l - cos(p);
the polar equation to the curve, called from its form the
Cardioid.
163. To find the equation to the epicycloid in terms of
the radius vector CP and the perpendicular on the tangent CY.
Produce CB to q; join PQ, Pq, then since for an instant
the revolving circle turns on Q, the motion of the point P
must be perpendicular to QP, and therefore in the direction
Pq : hence qP produced is a tangent to the curve.
Produce q P, and draw C F ± to it, and make
CP = r: CV=p; then •.• CV is parallel to PQ,
= : also, Cq = a + 2b ;
CQ' Cq'- '
160 EQUATIONS TO CURVES.
CoK. For the hypocycloid, c = a - 2b is T^ = -^; .-. AT = y -- - x.
AT dx dy
Cor. 3. These values may be derived from the general
equation ; for
dy
let x^= 0; .'. yi = AD = y - x — ,
y,= 0; .-. -x, = AT = y-^^-x',
or if these values be called y^ and x^, we have
dy dx
which are the parts cut oif from the axes of y and x by the
tangent.
I64f TANGENTS TO CURVES.
dti
Hence, having found ~ from the given equation to the
curve, AD and AT may be found ; then join TD^ it produced
is the tangent.
Cor. 4. If the axes be obh'que, we shall obtain similar
results, but — will not be the tangent of PTN, but the ratio
dx
of the sines of the angles the tangent makes with the axes of
X and y.
Cor. 5. The line NT is called the subtangent, and is
useful in drawing the tangent,
and NT = AN + AT = w + y , - x = y—.
dy dy
Hence to draw a tangent, find the value of NT. Join
P, T, and we have the tangent required.
Cor. 6. The length PT" of the tangent
= s/PN^ + NT^= \Jf + /-f. = y Vl + ^•
dy dyf-
167. Def. a line PG drawn from the point of contact
P, perpendicularly to the tangent, and meeting the axis in G,
is called the normal.
Since if y = mw + h is the equation to a line ;
.-. w = cc + 61 is that of a perpendicular to it ;
711
dy dy .
.-. since Wi= - cr. + y -
166 TANGENTS TO CURVES
chj
dy . "^ dw y-poo
.-. It m = — ; d =
Again, if Q be the angle between the tangent and a line
from the origin,
e = ta„-^-tan-'^;
X dx
.-. tan 6
y _^
X dx y — px
^ ^ ydy x + py
xdx
Cor. If we use differentials, multiply by dx.
._ ydx - xdy
and tan Q
'\/dx^ + dy^
ydx - xdy
xdx + ydy
170. It may be useful to collect these values in a table.
dy
(0 y\~ y - — (^1 ~ ''*^)> equation to tangent.
dx
(2) y\- y = (>^i — '^), equation to normal.
dy
(3) Subtangent NT = y~r .
dy
(4) Subnormal NG=y~.
^ ^ ^ dx
/ dx'
5) Tangent FT = y V 1 +-rT
dy~
(6) Normal PG=^y\/l + -^'
^ ^ dx-
y-poc dy
(7) Perpendicular on tangent = .— ^, ; where p= —.
11 — px
(8) Tangent APT = '—f—.
^ ^ ° x+py
TANGENTS TO CURVES. 167
dii
(9) JD = y, = i/-.v^.
(10) AT= -.v, = y^-.v;
dy
of these formulas, the first four are the most important to be
remembered.
171. Since the tangent of the angle which the tangent
makes with the axis of a? is — ; the ang-le at which the curve
dx ^
cuts the axis may be found.
For the angle which the tangent makes with the axis at
the point of section will be the same that the curve makes.
Find therefore the co-ordinates of the point of section, and
dy
substitute them in the expression for -^ , and the resulting
dx
value will be the tangent of the angle required.
Ex. 1. Let y = be the equation to the curve.
Here if ^r = 0, 2/ = ; and .-. the origin is the point of section,
^ dy 1 1 ,
and -— = = - , when x - Oi
dx (1 + xf 1
.-. tan = 1 = tan 45° ; .-. = 45°.
Ex. 2. Let the curve be the cycloid.
Here^=^^^^ = \/^-l,
dx ^y y
which is infinite if ?/ = ; or at the origin the curve cuts the
axis of X at an angle of 90'^,
Ex. 3. Let the curve be the circle, and the origin in the
centre.
dx y y- 2/
.r- or
y - px = y -i — = — , py + X = - X + X = 0;
y y
,\JY=^ = a; tan^Pr=-=co; .: z APT = QO".
a
168 TANGENTS TO CURVES.
172. To draw a tangent through a given point.
Let a and ft be the co-ordinates of the given point ;
x and y be the co-ordinates of the curve ;
dy
i.y\ - y) = — (-2^1-^) is the equation to the tangent.
dtV
But because it passes through a point where yi = ft and 0/^ = a;
from which, and the given equation to the curve, the point to
which the tangent is to be drawn may be found.
173. To draw a tangent parallel to a given line.
Let A = tangent of the angle which the given line makes
with the axis x ;
.-. — = A, since tangent and line are parallel;
and yi - y = A . (xi - w) is the equation required.
If it pass through a given point, the co-ordinates of the
point may be put for x^ and y^, and then from the given
equation to the curve, and from that of the tangent, the point
to which the tangent is to be drawn may be found.
174. To find the locus of the intersections of perpendi-
culars drawn from the origin upon the tangent, with the
tangent.
Let y=f{x) be the equation to the curve;
.'. y^ — y = — G^j -
■a - y
172 EXAMPLES.
.-. yy^- y^ = 2ax^ —2ax; .-. yy^ = 2a (xi + x).
CoE. Since yyi = ^a^x^ + x) is equation to tangent;
2a 2ax , 2a
.-. y, = — cTi + ; make — = m;
y y y
2ax y a
y 2 *w '
a
.'. 2/i =mx, + -;
m
an equation to the tangent which is often very convenient in
the solution of problems.
The equation to the normal is
y yx
^' 2a ^ 2a
— V if ^ o
Make — ^ = m^ ; .-. ^ = - = /»/;
2a 4a^ a
y^^ -K
.'. y +^^- = - 2amy - am{ ;
^ 2a
.'. 2/i = m^ {xi - 2a - am^)^
the equation to the normal in terms of its inclination to the
axis of X.
Ex. Two normals to a parabola intersect at right angles;
find the locus of their intersection.
.*. y + 2am = m{x — arn^) (1),
2a \ ( a\ ,^
and y = -- \x A (2);
m m \ m J
.-. 2a\m+—] = X im + —] - o ( m^ + -- ) ;
\ rn) \ mj \ my
.'. X - 3a = aim 1 (3).
a
But from (2) my - 2a = - x + --.
a
mi
1
(l) — « + 2a=ir— awi
^ m
2.
EXAMPLES.
173
■-■y'ih-'") W'
.'. y^ = a^ i mj = a{x - 3d),
the equation to a parabola.
(2) Find the equation to the tangent in the eUipse ;
The centre being the origin,
V" a'
dy b^ a?
dec a^ y
••• yvx - 2/ = -T-'^<^i + -2 -^ = - -2 ^^i + f> -y i
add
Let yi = 0;
(B CN
.'. C T X CN= CA\ (See Conic Sections.)
and NT = CT - CN = — "^ = sub-tangent,
or iVr X CiV = (a + c^) (a - v) = ^^iV x AN.
174 EXAMPLES.
b . (a — .r)
Cor. 1. Make x^ = a; .-. y^ = AD
Xi = — a; .'. yi = ^,Z>
h. {a + a)
.-. AD.A,D, = b'= CB\
dy b'' b^
also NG ^ - y — = - a; = - CN.
a X a a
Cor. 2. The equation to the tangent may be written
a^yy^ + ^^xx^ = arb^ ;
b^x b-
••• 2/1 = - -2- -^i + - •
ay y
b'x , 3 b'cc'
Let »w = ; .*. ma' = -j-o;
a-y af
... m^a- + 6' = W
6'a?' + a'2/^ a' 6* 6
a^2/^
•'• y\ = ^^-^i + \/m^a^ + &^ ,
an equation to the tangent in terms of the angle it makes
with the axis of x.
Ex. Find the locus of the intersection of pairs of tangents
to an ellipse at right angles to each other.
y = mx + \/ rr^c^ + 6^ one tangent ;
... 2/ = - - + V — +
&", the other;
.*. y — mx = y/rr^c? + 6^^,
my + Ty
«i=0; .-. JT = a^ +
Vy
.-. ^D + ^T = cr + 2 \/.r2/ + !/ = (n/-^ + v/?/)' = '^•
(7) Draw a tangent to the cycloid.
AN=x, NP=y, AB = 2a.
EXAMPLES.
177
Then
dy V 2aa? - x'^
dx
d.v
y .X
dy V 2aa? - x^ '
_ NP .AN
NT : NP :: AN : NQ;
and ziVis common to As JNQ, TPN \ .'. they are similar,
and Z PTN = i QJN; .-. the tangent TP is parallel to the
chord AQ.
Also since z. AQB h always = 90, PG is parallel to BQ.
(8) Draw an asymptote to the hyperbola.
b / H b I 2a\^
b / 5 b ( 2a\
= ± -\/2ax + .t'' = ±- 07 1 + —
a a \ X J
= :t
1 .2
6 5 1 a^ 5
= ^ - .\x + a--.- + — + &c.
a 2 X x^
and therefore y = ± - (,r + a) is the equation to two asymptotes;
a
and since if .r = 0, y = ^ b; and if ?/ = 0, <»? = - a, both will
pass through the centre, and be equally inclined to the axis of x.
(9) Draw the asymptote to the curve.
y^ =z x^+ a x^ = .iM 1 + - j ;
, a\^ , a A B ^
.'. y = x [l +-] = .t? U + i . - + - + — + &c.
a A B
X + - + - + — + kc.
3 X x^
'. y = X + - is the equation to the asymptote which cuts the
axis of X at an z = 45", and at a point x
12
178 EXAMPLES.
(10) Let y . {ax + /r') = w^, draw the asymptotes.
x^ 1 .r^ , . ir
y = = by putting - = e
ax + b- a X + c ' a
a c a X X x"
1 +-
x~ ex c- c
= + + &c. ;
a a a ax
.'. ay = .7?"- ex + c^ is the equation to the asymptotic curve.
This being put under the form
ay - fc^ = w - ex ■] = Ix —
(--i)'="(!'-f°^)
shews that the curve is a parabola, the axis of which is per-
pendicular to the axis of x, and the vertex determined by
c e~
making x^ = - and ?/j = J— ; the latus rectum = a.
The curve has also a rectilinear asymptote parallel to the
axis of y ; for making x = - e = , y becomes infinite, and
a
-~- is also infinite ; therefore a line drawn perpendicularly to
ax
the axis of x at a distance from the origin, will when pro-
duced to infinity be a tangent to the curve.
178. When the equation between y and x cannot be
solved with regard to y, it is frequently very convenient to
substitute zx for y ; and then from the given equation to find
X and y in terms of z ; and z being so assumed as to make
X and y infinite, the inclination of the asymptote, if any, to
y
the axis of x is known (for z — - = tan inclination) ; and JD
X
and AT being determined, the position of the asymptote may
also be determined.
(11) Let ay - bx'^ + e^xy = 0; make y = xz%
.-. a.r'^' - h.v'^ -\- c^arz = ;
ASYMPTOTES.
179
•. x"-
c'z
h - az^
= w ; if az' = h :
; or
z=V^l
a
•' y
a
is the equation to
the
asymptote,
since
AD
= y-px-
= -;"T="^
ify
= co; and ;
r = oo,
y 2c^xy
and AT = X - - = — = ; if 1/ = cc; and a; = co.
p A-ay + ex
(12) Find when the curve, which is the locus of the
general equation of the second order, has an asymptote.
Here Ay~ + Bxy + Cx- + Dy + Ex + F = 0,
or y' + 2 {ax + b) y + cv' + ex + f = 0,
dividing by A, and making the proper substitutions ;
.-. y-+2{ax + b)y + {ax + by = {a^ - c) x^ + {2ab - e)x + b'-f,
and y=- {ax + 6) ± \/(a^ - c)x^+ {2ab - e) x + {b' -/) ;
f( ^^a/2 2ab-e b'-f\
f b /— / 1 2ab - e A B\)
= -a?{a+ - ± \/a^- cil + — — ^ + -|. _ U
[ X \ 2x {a~ - c) x^ XV )
2ffi6 e A
= - \ax + b =i= \/a^ - c {x + J ^ — — + T "*" ^^•) \ '"
/—, , /, 2ab - e \
y = - {a^ x/a' - c)x- U ± .-- ,
V 2\/a^ - cl
a ~ c X
and therefore the equation to the asymptote is
2ab
7?" (^
which is possible when a^> c, or > -,-> or B^ - 4-AC >0,
4 J- A
or when the curve is the hyperbola.
CoR. If a^ = c, the curve is the parabola, and the equa-
tion being of the form y = - {ax + 6) =*= y/ mx . \/ 1 + —
X
cannot be reduced to the form y = Ax + B.
Prob. Find that tangent line to a given curve which cuts
off from the co-ordinate axes the greatest area.
n2
180 ASYMPTOTES.
Area = ^^o2/o= i(y - P-^) ('^ - -^
{y - p.vf dy
^ ; where « = —-;
^ p ax
(y - pxy
— — - a maximum ;
du 2 ^ ^ fdy dp^ {y - px) dp _
P
lu 2 (dy (lp\
dx p • ^ ^ \dx ^ dxj p" dx
.'. - 2px - (y - px) = ;
2 '*
P )
Therefore also y = ^y<)\ and the tangent is bisected at the
point of contact.
Cor. From this it appears that the least polygon of a
given number of sides which will circumscribe a given oval,
must be such that every side of the polygon must be bisected
at the point of its contact with the oval, since the interior area
must be least, when the triangular spaces as in the preceding
problem are greatest.
In the problem the axes are supposed rectangular, but
if they be oblique, the same results are obtained, but area
= 1 ,ro2/o ^ sine of inclination of axes.
EXAMPLES.
(1) Let ?/" = rA"-i.r; NT = nx; NG = ~
a
If w = 2, the curve is the parabola ; NT = 2x, NG = -
(2) Let the curve be the Witch
2ax — X 4«.
NT= ; NG =
o x~
(3) The focus of a parabola is in the centre of a given
circle, its vertex bisects the radius; find the point and angle of
intersection of circle and parabola.
EXAMPLES. 181
(4) Shew that the curve defined by y^ = 4a a-, intersects
4
the curve defined by / = {co - 2rt)% at a point where
27a
X = 8 a, and find the angle of intersection.
(5) If 2/^^ = 4 o (.r + cf) be the equation to a parabola,
the origin in the focus, shew that the points of intersection of
the tangents, and perpendiculars from the focus, are deter-
mined by the equations
y
,r, = - a, and 2/i = - •
(6) The locus of the intersections of tangents to the
parabola with perpendiculars from the vertex, is the cissoid.
(7) Find the length of the perpendicular drawn from the
focus of an ellipse upon the tangent, and shew that the locus
of their intersection is a circle, radius = a.
(8) Given two points A and B, find the locus of P when
the angle PBA is double of the angle PAB, and draw an
asymptote to the curve traced by P.
(9) Draw an asymptote to the curve defined by y^ + o)^
= 3a.vy, and determine the points where the tangents are
parallel to the two axes.
(10) Find the point and angle at which the curve
3/ = 0? (a? + 2)^ cuts the axis. At the origin, angle = 90.
(11) Find the same when y - 2 = (x - 1) y/x - 2, and
the values of x and y when the tangent is perpendicular to the
axis of X. (1) 2/ = ; cr = 3. (2) ?/ = 2 ; a? = 2.
(12) If y'^-=SaP-w^ draw an asymptote to the curve;
find its greatest ordinate; and the angles the curve makes
with the axis of x at the points, x = 0, a? = 2 and x = 3.
Asymptote y = - x + \\ ordinate = 2 ; angles are 90, 0, 90.
x^ + ax^ + a3
(13) Draw the asymptotes, (1) when y = 2 _ 2 — ,
(2) when y^ = ax* + mx^, and (3) when y = a*.
(14) If in the ellipse, = z CPG, and I = the angle PG
^ tan /(a' -6')
makes with the axis major ; tan = —5 — r^- — :rr ■
'' a +b^ tan- 1
. y ■ o,
(15) Shew that the curve whose equation is - = sin-;
182 EXAMPLES.
where j/^ - 2ra' + .2?*= ; intersects the axis of x at points
2a
determined by w = — , n being any whole number.
nir
(16) The normal to the curve defined by y^ = i^ax, is a
4
tangent to the curve in which y^ = . (,r - 2 ay.
(17) Draw a tangent to a circle, cutting the axis of ,v
at 30°.
(18) In the conchoid, where aPy^ = (a + .vf . (Ir — os^),
ATT- - -^•(« + ^)-(ft'--^')
x'+ab'
Before differentiating take the logarithm of the equation,
(19) Draw a tangent to x^ + ys = a^ ; shew that the part
of the tangent intercepted between the axes = a, and that per-
pendicular on tangent = \/aaiy.
(20). The centre of an ellipse is the vertex of a para-
bola, the major axis of ellipse is perpendicular to the axis of
parabola, and the curves intersect at right angles, prove that
major axis : minor axis :: y/^ : 1.
(21) If PY and QF be respectively tangents to corre-
sponding arcs of a cycloid and the generating circle, shew that
the locus of Y is the involute of the circle.
(22) Find the angle of intersection of a rectangular
hyperbola and a circle having the same centre, radius of circle
being 2 a ;
angle = sin"' ^\/l5.
(23) If TP and TQ be tangents to a parabola, and S
be the focus, shew that
SP.SQ^ST'.
(24) If 2y = c(e^ + e~^), (the equation to the catenary),
the normal = - y~.
(25) If — = — + — be the equation to a curve, shew
y" a." w"
that n" .AT = .v"^K
EXAMPLES. 183
(26) Find that point in a parabola, at which a line from
the vertex makes the greatest angle with the curve.
Ans. X = 4 a.
(27) If A be the extremity of the diameter of a circle,
FT a tangent, PN an ordinate, and AP a chord, prove that
JP bisects the angle TPN.
(28) If y" - (a + b.v) y"-^ + (c + ex +fx~) y"-'^ - &c. = 0,
be the equation to a curve of n dimensions, prove that the
sum of the ordinates divided by their respective subtangents,
is a constant quantity.
(29) If C be the centre of a circle, AQ a. chord, and let
CNR be drawn cutting ^Q in N, draw NP perpendicular
to .^Q and = NR ; find the locus of P and draw its asymptote.
If 2c be the length of chord, a = radius of circle, origin
centre of chord, and the chord be the axis of x, then
(y - a)' = x^ + a~ - c'.
(30) ABD is a semicircle, centre C, and diameter AD,
EF is a chord parallel to AD, CQR a radius cutting EF
in Q, bisect QR in P; find the locus of P and the position
of the asymptote. The curve is the conchoid.
CHAPTER XII.
THE DIFFERENTIALS OF THE AREAS AND LENGTHS OF
CURVES: OF THE SURFACES AND VOLUMES OF
SOLIDS OF REVOLUTION: SPIRALS.
179. One of the applications of the Integral Calculus
is to find the areas of curves included between given ordinates,
the lengths of their arcs, and the surfaces and contents of
solids.
The solids of which we shall treat are called solids of
revolution, since they may be supposed to be generated by
the revolution of a plane figure round a line, thus termed an
axis. Hence it follows that every section perpendicular to
the axis will be a circle, the radius of which is the revolving
ordinate, and every section made by a plane passing through
the axis will reproduce the original area.
Considering the areas and lengths of curves, and the
contents and surfaces of solids, to be functions of one of the
quantities w or y, we can, by the Diff'erential Calculus, find
equations between the differential coefficients of these functions,
and expressions containing x or y, by which we shall hereafter
obtain the values of the functions themselves.
We shall find it useful first to establish the truth of the
following Proposition.
180. If J + Bw, Jj + B^A, and A + B.^os be three alge-
braical expressions taken in order of magnitude, viz., the first
greater than the second, and the second greater than the third,
then shall
Jj = A, whatever be the value of w.
For {A + Bx) - (J + B,a!) is > (^ + Bx) - {A^ + B^x);
A + Bx
.-. if (A + Bx) -{A + B,x) = 0, or ^ ^ ^ ^^ = l ;
.„ A + Bx
.-. a fortiori will —— = 1.
A^ + B^x
A+Bx .A
But as X decreases -— approaches - or 1 ; and
A + B,iX A
when X is diminished without limit it actually equals unity,
and
DIFFERENTIAL OF THE AREA.
185
— which becomes -—
bv the continued diminution of a?, also is equal to unity
.-. — = 1, or A = A.
A,
AREA OF A CURVE.
181. Let AP be a curve, y=f{x),
the equation to it, where AN = cc,
NP = y ; and let A = area ANP.
dA
Then =2/.
dx
Let iViV, = h. Complete the paral-
lelograms QNx and PN^.
Then the area P.PNN, is > □ PN,, < ED QiV, ...(1).
Now A depends upon w ; for as tc changes, A changes ;
.-. A = ANP = (p (x) ; and .-. AN,P, = (p (x + h) ;
dA d~ A K-
dx dx- 1 . 2
and □ PN^ = yh.
□ QN, = hxP.N, = h .f{x + h) = h{y A-ph +Ph'\ ; P = ^ ;
therefore, dividing by h, we have by (l),
dA d^A h I ni.2
dx dx^ 1 . 2
dA d'A h ^
i. e. y + ph + Plv, "T" + T^ ,~^ + ^^- ^"^ V
^ ^ dx dx^ 1 . 2
are in order of magnitude; whence, by the Lemma,
PP,N,N = cj){x + h)-(p (v) = -^" A + ::^ -_ + &c.
dy
dA
dx
y-
186
LENGTH OF A CURVE.
LENGTH OF A CURVE.
182. If s = length of the
curve JP ;
ds / dy'^
Draw tangent PM,
and chord PP^ .
Then arc PP, > chord PP, < PM + MP, .
BntarcPP,^AP,-AP=ch(x + h)-cf)(x)='^h + ^-^ + kc.
1 rv ^ ^^ ^ dx dx' 1.2
chord PP, = y/Pm^ + (P,m)2* = ,/fi^ + {ph + PKy
= h v/(l + f) + 2Pp A + P^ AS
PM = \/Ri^F+Mn^* = x/a^ + pVi^ = /t \/l + p%
i¥Pi = MN,-N,P, = (y+ph)- (y+ph + P/i^) = -Ph' ;
whence, dividing by h,
ds d^s h I ■ — /
-T- + -— . + &c. > \/l + p'+ 2 P/>/i + P'h^<\/l +p^- Ph
dx dor 1.2
, Pp ,—
\/l+p
dx ^ (/.r'
VOLUME OF SOLID.
183. If V be the volume of a solid of revolution APp^
dV
dx
■y~.
Mm= I'm . tan M I'm- h . tan I'TN =^h . '-^ .
VOLUME OF SOLID.
187
Let AN
NP = v\: .: AP,p,=f(.v + h) = V+~h +
dV. d'V W
,-._,., + &c.
ax dx'^ 1 . 2
NN, = li\
Then the solid PppiPi is > cylinder PMrn^p,
< cylinder RP^p^r ;
. dV ^ ^V W ^ ,.
dx' dx~ 1 . 2
<'K{y + ph + Phyh,
dV cPV h ^ 2 . , ^ „ „
or — + -— 2 — - + &c. >7ry^<'7r(y +ph + Ph~)\
dx dx 1 . 2
or > iry~ < iry^ + ^irpyh + &c.
whence — - = ttw .
dx
Prob. The surface of a truncated cone, of which the
radii of the greater and smaller ends are a, 6, and the slant
side s, -Trs{a + h).
Let I = length of cone, radius of the base = a,
l,= =ft;
therefore, surface of frustum
=: irla - Trlih = TT \sa + l^{a - ft)|,
but ^ or ^, + s : /j :: a : h ;
.' . s : I] :: a - h : h ;
.-. sh = /, (a - b) ;
.-. surface of frustum = tt.v . (a + b).
188
SURFACE OF SOLID.
SURFACE OF SOLID.
184. If ^ = surface of the solid of revolution APp,
dS , / dy"
Draw the tangent PM^ and chord PP^.
AN = w
NP =y
AP =
NNi = AJ
Then, surface generated by arc PP, , will be
> than that by chord PP, , < by PM and MP^ .
Now chords PPi and PM generate truncated cones, of
which the surfaces respectively are
7r{PiV+P,iV,}PPi, and tt {PN + MN,\ PM;
and MPi will generate a circular zone = tt {MNi^ - N^P^') ;
and the surface generated by arc PPi
dS d'S h~
del? dx' 1 . 2
But {PN ^P,N,]PP,
= {2y + p/i + F li').\/)f
(ph+ Pli'f
= (2y + ph-\- Ph") A V I + F + Mh\
'here M h = terms involving h ;
and {PN + MN,) PM
= (2y + ph) . \/A- + p^'h^ = (2y/ + p/i) li \/l + p^
SURFACE OF SOLID. 189
also MN'-N,P;'
= (y+ phf -{y + ph + PhJ = - Ph' (21/ + ^ph + Ph')
= _ ]S^h', by substitution ;
... 1^ + ^ i-+ fee. > TT (21/ + pA + Ph') y/^ + p^+ Mh
< 7r{2y + ph) x/l + p^ - TrNh
>27ry'\/l + p^+Mh + terms involving h,
<27r?/V 1 + p^+ irph's/l + p'^ - -rrNh
yMh
>2iry\/l + p' + TT / „ + &c.
Vl + p^
<2 iryy/l + p~ + Trph \/l + ]? - rrNh ;
dS
/ o 1 / dy' ^*
-—=27ry\/l+p' = 27ry\/ l + --^= 27ry --.
dx dx^ dx
Cor. 1. Hence it appears, that if in a curve ANP,
A = area of ANP; and SA be the differential of the area, and
^.1; the differential of the abscissa ; Sx being very small, that
U dA .^ ,
= =2/; .'. dA = y6x,
dx dx
or the differential of the area of a curve is equal to the
rectangle of the ordinate, and the increment of the correspond-
ing abscissa.
Cor. 2. If Ss be the increment of the arc AP; corre-
sponding to the increments ^x and ^y of x and y ;
. Ss ds
then smce t^- = -;— ;
dx dx
)x dx'' oar
.-. h = ^/Sf+Sx';
whence it follows that the increment of the arc is the hypothe-
nuse of a right-angled triangle, of which Sx and ^y are the
other two sides.
Hence also it appears that the chord PPi= arc PPj ulti-
mately.
190 SURFACE OF SOLID.
Cor. 3. If a be the angle which the tangent makes with x ;
.-. ^^ = sec a ; cs = dx . sec a-
Cor. 4. In the same manner, if SV and ^^S" be the dif-
ferentials of the volume F, and of the surface S of a solid,
dx
^ ^S ^ / df cs
and -^ = 27riy. V 1 + :r^ = Stt?/. y- ;
dx (ix~ ex
from the former of which expressions it follows that the diffe-
rential of a solid is a cylinder, base Tr?/^ and altitude ^x: and
that the differential of a surface is the convex surface of
a cylinder, the circumference of whose base is Stt?/, and alti-
tude ^s.
185. The expressions just obtained, and those of the
preceding Chapter, are only applicable, when the equation to
the curve is known in terms of the rectangular co-ordinates;
we shall now find corresponding expressions for the perpen-
dicular upon the tangent, the area and length of a curve,
when referred to polar co-ordinates; that is, when r =/(0),
or p =/(r), p being the perpendicular on the tangent, r the
radius vector, and the angle traced out by r.
186. To find the expression for the perpendicular on tlie
tangent in polar curves.
SN = X, SP = r,
NP = y, SY = p, and
Art. U)(), SY
But X = SP cos PSN =
y = SP sin PSN =
SPIRALS. 191
dy _ ldy\ lde\ _ dB
''" d7v ~ KdQj KJx) ~ dZ '
dO
and — ; = r cosO + sin 9 .-rr-;
dO dO
.-. .y j^ - ^^ V5 = »• sin'' + r cos'' = r ,
whence having given 6 = f(r), p may be found in terms of r,
or of r and 6; but the formula may be put under a more
convenient form for practice ;
„ dr'
1
r'
"^ rf6>'^ 1 1
dr'
'dO'
•
Let r= -;
u
•'•
1 dw
" = r^ -'-de
1 2 ^W'
1 rfr
?'de'''
I87. To find the differential of the area.
dA
Let J = area ^iVP, then will — — = ^r'.
du
For ASP = ANP - SNP = ANP - — .
2
192
Cor. 1
SPIRALS.
^ d.ANP d.(ANP)
''"' d. =^^ ae '
dx
dA dx . [ dx
■■■le'^-Te-^-V'de^
de)
M^fe-^:)
= ir^(Art. 186).
. If e=f{r), to find ^.
^.111 dr^
• • d&' [f r'j ' p'
de
' dr r s/t^ - p^
a useful equation in dynamics.
188. To find the differential of the length of a spiral.
Let s = AP; and let y and x be the co-ordinates of P;
ds / dy^
but s, .r.', y being functions of 0,
ds ds dd dy dy dO
dx dd' dx^ dx dO dx
" de ^ de' de' ^ de^'
189. To find the same, if s=f{r).
ds ds de / ,rf6>'
dr de dr ^ dr'
= V.-^-
r' - f \/r'
p-
CoK. Hence cos SPY = ^ ''' ^' = —
r ds
SPIRALS. 193
190. To find the tangent of the angle whicli SP makes
with the curve.
PV ^r' -p' dr'
\X r = -\ = - u . -— .
Ill du
191. To draw a tangent to a spiral.
P the point to which the tangent is to be
drawn. S the pole. Join SP. Suppose PT
to be the tangent. Draw SY A. PT, and ST
±PS, then ST is called the sub-tangent. ^V.^
And^7^ = ^^P.f?,= -.Z^=alsor^ ^^
PY ^/r'-p" dr
Find therefore from the equation to the spiral , or
V r^ - p^
de .
r^. — , according as the equation is p =f{r)^ or Q=f(r).
dr
Draw ST perpendicular to SP and equal to either of these
values. Join T'P, it is the tangent.
r. c- c.r„ , d^ dO 1 /duy
Cor. Since ^^ = ±r^—-= =f -—;,.•.-—.= -— ;
dr du ST^ \d 9 1
1 , /du\^ 1 1
■'* SY' ^ '^ \del SP"'^ ST"'
192. Asymptotes to Spirals.
If ST remain finite when SP is infinite, a tangent may be
drawn which will touch the curve at a point infinitely distant
from S, and is therefore an asymptote. And since those lines
are said to be parallel which coincide only at an infinite dis-
tance ; the asymptote must be drawn parallel to the infinite
line SP.
. dO
Hence to construct; find and r'' . — when r is infinite.
dr
Draw SP at the angle thus found, ST perpendicular to SP,
and TP parallel to the infinite radius vector, TP produced
is the asymptote.
193. Asymptotic circle.
If in the equation B=f(r), 9 becomes infinite when
r = a ; but impossible if r be > a ; then if we describe a circle
13 O
194
SPIRALS.
with radius a, the spiral will make an infinite number of revo-
lutions within the circle, and constantly approach the circum-
ference, without exactly reaching it. In this case, the circle
is called an exterior asymptotic circle. But if r = a make d
infinite, and r < a, make d impossible, the revolutions of the
spiral will be without the circle to which it is always tending ;
and the circle is an interior asymptote.
Ex. 1. e = \og
a - r
a + r
Ex. 2. e = log
\r + a
194. The preceding expressions
may be conveniently obtained by a
direct investigation.
Let area JSP = A ; SP = r,
AjSP = e; SY=^p.
Draw SQ very near to SP,
and draw PT±SQ.
Then area PSQ = ^A,
^PSQ = ^e: QT=^r;
now PT is ultimately = a circular arc ;
.-. ^A = ISQx PT very nearly
= i(r + ^r) .rS9 very nearly
=
V
^9 +
ir,
^A
V
+ ir^r.
But as Q approaches P,
Ir continually diminishes, and
^ U dA
ultimately vanishes ; but then T5 = T5^ '
^e de
and
dA
de
2' •
195. Again (chord PQY = PT' + QT"- =r'.^e' + (Irf
(chord PQ)^ _ ^ _ . (^r\'
•■• w ~ sff'~ Uei'
d,s / dr-
.whence -r^ = \/ 1 + -j^ •
dO
de'
195
Also from similar triangles, SPY and PQT,
PT _SY ^_ _j>
Qf'PV' ''h-'^r^^'
dd_ p
■ ' dr r y/r^ - f
PQ _^8 _SP _ V
Qf~h'~PY~ ^?'Zf '
ds r
.(1).
And
dr ^,»-f'
PT
196. Since tan SPY = tan PQT = —_ ;
0,1
se do
.-. tan SPY = r . ^ = r . — ,
dr dr
PT ^e de
sin SPY = — = r . ^ = r . -- ,
PQ h ds
QT ^r dr
Also SY : SP :: PT : PQ;
PT „ S9 , d9
■''P = '-pQ='"'Ts = '"-d-s-
It is necessary that the student should be familiar with
these methods of obtaining the above results from the triangle
SPY, and the ultimate rectilinear triangle PQT; as he will
find their use in dynamics and in the early sections of Newton''s
Principia.
197- Prob. To find the equation to the curve which is
the locus of the intersection of the tangent and the perpendi-
cular from the pole ; or to find the locus of Y.
SP = r ; SY = p, and let p, be the perpendicular from S
on the tangent to the curve traced by Y.
Let = z ASP; = z ASY. See fig. Art. 194;
ry/r-p^ PVP -Pi
But (h = 9 - PSY =e - cos-> . ? ;
^ r
02
196
SPIRALS.
p,dp
pdr
rdp -
T\/?
pdr
P Vf - Px
rVt^-f
-f
rdp
r ^/r2 - p2
P^
P .
r-
v^-P.'
-Vr^-f'
p
••• P.'-
= — ; and p, =
•>
r
which is the value of the perpendicular on the tangent in the
required curve : and if r^ be put for p; and since r is a func-
tion of p =/(ri),
••• Pifi^i) = rA
will be the equation to the locus required.
Ex. Let the spiral be the equiangular where p = mr ;
r{- mr^
.'. r, = mr ; •• Pi = — = = wr,,
r r^
an equation to a similar spiral.
EXAMPLES.
Example. Find the value of p in the Conic Sections.
m
r =
where m = ^ latus rectum
1 + e cos "^
1 e ^ du e . ^
.'. M = — + — . cos ; -— = . sin ;
mm dv m
du~ 1 ( , „ ^ o )
= — (2mM _ 1 + e^^
1 f2m-r(l-e^]
"^*1 r J'
2 wi' . r
••• P "
2m -r(l - e'O
EXAMPLES.
19'
mr
(1) In parabola, e = 1 ; .-. f = -- , and m =
= 2^^;
.-. SY^-^SP.SA.
(2) In ellipse, e\; ^^ ~ ^ = ^ '
. .^...^^'^^l— = ^^^ :
2m + r(/-\) 2a + r
g BC'.SP
and therefore in ellipse and hyperbola, SY = — j— — .
o"
(4) Find the equation between p and r, when ^ = —
r
=
a"-
z<" ; .
•. w"
9 ^
du
1
=
6"
by
substitution
dd
na" .u
Ti-l
dv?
1
,.2«-.
1
f6'" + r"l
. u-
^de^-
7^
+
6^"
6»
. r
i 6'. J'
.-. p =
(5) Draw a tangent and asymptote to the spiral ; where
Q^^^azc- .■ -^=— = -; .: ST = a; or the locus
r ' ST dd a'
of T is a circle radius = a.
Since aST is constant, and 6 = ,^| --
when r = CO. Produce SA indefi-
nitely. Draw ST ± SA and = a.
Then a line from T parallel to SP
will be the asymptote required.
(6) If r-= a'^ cos 20 which is the polar equation to the
Lemniscata ; find the equation between p and r.
198 EXAMPLES.
Here u' = -^ ,. ; cos 2 = ---. ;
a^ cos 2 71^ a'
— = ——, -; /. -— ^ w^a^sin20;
du u-^a-s\n2d ad
and sin 20 = V 1 - 4^4 = ^^ '->
du
•'* dd
= u
s/a^
'u'
-''
li'
(duY
= u'
+ a'
u'
- u'- =
a'
u'' ;
1
a'
7'
p
(7) Let r = a^, the equation to the logarithmic spiral ;
dr ^ . ^ dO p 1
dd dr r\/r^-p^ Ar
.-. y-^ — „ = — or — — = tan SPY = - ;
s/r'-p'' A PY A
that is, jL spy is constant, and on this account the curve is
called the equiangular spiral.
Cor. 1. Since -^ ^ = A ; r.-^l+A';
p p^
.• ^ = sin SPY ^
\/l + A'
r
.'. p = — = mr, by substitution.
Vl + A'^
Cor. 2. The radii including equal angles are propor-
tional.
Let SP and SPi include an z a,
and SQ and SQi include the same angle.
Let z ASP = e, and z ASQ = (p ;
.-. SP = a\ SQ = a*^,
^^A = «'"'% .SQ, = a*+";
• '. «„ = « , and = n ;
.S'P ' SQ
EXAMPLES. 199
••• ^ = I? ' "r SF : SP, :: SQ : SQ,.
Cor. S. Given the ratio of SP and SP,, which include an
angle a; find a.
Let SP : SP, :: \ : I + c.
But ,S'P= a^ and 5'/^ = «'+";
.-. =! + <■ = «";
... w(i + c) = a^ = — %; ifi3 = /6^pr,
^ ^ tan /3
or a = tan /3 . log (l + c).
(8) If >• = a (1 + COS0) ; find the equation between
p and r. Ans. p' = — .
(9) Find the equations between p and r ; (l) when
r = a (6^+^ + e""^) ; (2) when r = a sec nO.
r or
(10) In the ellipse, if p be the perpendicular from the
centre on the tangent, and r = distance of point from the
centre, prove that
CHAPTER XIII.
SINGULAR POINTS IN CURVES.
198. If in the equation to a curve expressed by y =/(,r),
where y is the ordinate, and x the abscissa ; some value of .x-
as a makes any of the differential coefficients 0, - , or - ,
^
the point so determined is called a singular point.
(l) Let the values of the first differential coefficient be
considered. .
dy
Since —^ represents the tangent of the angle which the
dx
tangent makes with the axis of 1
^
therefore the deflexion from the tangent, or J/P,
(fy h" d^y I
d^Tr~2~ dx^ 2
d^y ¥ d^y h?
d^y h" d'y h'
in figure (1) ^ N.M - N,P. ^ - ^ - - ^ —
in figure (2) = N,P,- N,M= +
dx- 1 . 2 dai' 2 . 3
-&c.
+ &c.
and since li' is positive, and that h may be taken so small,
that the first term of the expansion may be made greater than
the sum of all the terms that follow it, the algebraical sign
d'y
of MPi will depend upon that of -— .
Therefore when the curve is concave to the axis,
MP. = — &c. ; and when convex to the axis, it
dx' 2
202 POINTS OF CONTRARY FLEXURE.
= + —,, \r &c. Hence, {y being positive,) a curve is
1 • T ^'.V . . .
convex or concave to the axis, according as — '- is positive or
negative, or generally according as y and ~ have the same or
different signs.
CoR. If we suppose PT to be drawn _L to PiNi, PT = h,
, MP' , d-y d'y h
and = == i . — - :p — ^^ . J- &c.
PT' ^ ^ dw'^ d.v^ 2.3
and if h be constantly diminished, the limit of the ratio of
MP' : PT' will be = ± 1 . ^ .
-^ d.v'
Hence, ultimately, the deflexion from the tangent,
d'v
200. Sometimes the curve after being convex to the
axis suddenly changes its curvature, and becomes concave,
the point at which the change takes place is called a point
of inflexion, or of contrary Jlexure. .
If the tangent at this point be produced, one branch of
the curve will be above, and the other below it, consequently
on one side of the point in question -— -, will be positive, and
d'y
on the other side, negative. Hence at the point itself — —
dx^
must = 0, or 05, for no quantity can change its sign without
passing through zero or infinity.
There is not however a point of inflexion corresponding
d~y
to every value of x, that makes --— s = 0, for not only must
dx'
this equation be satisfied, but — -^ must change its sign after
dx
having passed through the point under consideration.
d~ y
Also if the same value of x that makes — ^„ = also makes
dx~
— ^ = 0, tlicre may not be a j)oint of contrary flexure.
dar
POINTS OF CONTRARY FLEXURE. 203
For since — ^ is a function of w, write x + h and .v-h for x,
dx'
and then — - becomes, on these two suppositions, either
dco-
d-i, d'y dUjK' ^
•^ ^ dw' dx" dx^ "2
d'y d'y d'yJi' ^
But at a point of inflexion -r4 = ^ ^ •"• ^^^ deflexions
^ dx^
from the tangent at points x + h and x - /i are respectively
proportional to
+ -74 ^i + t4 - + &c., and - — ^ /^ + -— - - &c.,
d^y -f ^'y _
which have contrary signs if --^ do not = ; but if — — = 0,
.4
and — does not vanish, the deflexions before and after the
dx^
point will have the same algebraical sign, and the branches are
both concave, or both convex, to the axis.
And hence in general there may be a point of contrary
flexure, when the first differential coefficient which does not
vanish is of an odd order.
Hence, to find whether a curve has a point of inflexion,
put — = 0, or - , and if a be one of the values of x so
^ dx'
determined, substitute a + h, and a- h for x in the expres-
sion for — . Then if —4 be affected with diff*erent signs,
dx' dx^
X = a gives a point of contrary flexure.
Ex. 1. The cubical parabola ay = x^.
x^
y = —; and if x = 0, y = 0,
a
dy Sx^
dx c?
d"y Qx
Jx'^ ^'
204 POINTS OF INFLEXION.
If X be positive,' or negative, y and — -^ are positive or
negative ; the curve is therefore always convex to the axis.
If.r = 0, ^, = 0.
It X = n, — ^ = -^ IS positive.
It x = - n, -- = IS negative.
da- a-
The origin is therefore a point of contrary flexure ; also,
? .r = 0, makes
gent to the curve.
dy
since .v = 0, makes — = and v = ; the axis of /3
Ex, 3. Find the point of contrary flexure in the trochoid.
t/ = a (1 - e COS0) ; X = a{9 - e sin 9) ;
dy . ^ dx , ^_
.-. — ^ = aesin^; -— = « (l - ccos0);
dd dd
dy esin0
' ' dx 1 - e cos
SPIRALS.
•205
(Py e cos (1 - e cos 6) - e s m^6 dd^
rf^ ~ (1 - e cos 9)'^ ' d.v
e cos 9 - e' 1 _
^(l-ecos(9)- a(l-ecos^)
e (cos 9 - e) .„ ,.
= — ^^ —=0, if cosO = e;
a (1 - e cos 9y
e;
■. cos = e gives a point of contrary flexure,
b-\ a-
and y= a(l - e') = ail - -^ 1 =
.2 6'
crv a
201. Points of contrary flexure in spirals.
Let there be two spirals, one concave and the other convex
to the pole. Take two points P and P, in each near to each
other, and draw SV and ^STiX to the tangents at P and P,.
Let SY=p, SP = r, and SP, = r + h, and p =f{r) ;
therefore if A be the difference between SY^ and SY, we
have in figure (l), where the curve is concave to the pole,
dp d'p h~
A=/0- + A)-/(r)=-^^.y> + ^— +&C.;
and in figure (2), where the spiral is convex to S,
dp d^p h
dp
- &c.;
and as h may be taken so small that — h may be greater
than all the terms that follow, we see that the spiral is con-
dp .
cave or convex to S, according as — - is positive or negative.
Note. PiVi in the figure should be a straight 11
206 SPIRALS.
Hence at a point of contrary flexure -7— = O-* find changes
its sign immediately before and after the point.
]
EXA3IPLE.
Let r = a 9",
, find
the
point
of contrary flexure.
d9
1
9
vl ^ ■
dr
naO"
"'
nr
7ir -
P
; Art,
1 S7 Cor 1 •
dr r
x/r^-
^^"
1
22 2
P
I
' •'
^ _^ w^a" ?•" + w-ff"
' y/r"--p^
r" r"
••• P
w+l
r
2
» 4 i
2 "■
dp n +
r
2
n+l J.- -;p
dr n
'' /2 2 ^'
71 \/V'+ w''a"
omitting the denominator ;
12 ^3
.-. (n + 1) 7'"(r" + w;^«") - r" = ;
i ? -
.-. r" |wr" + ri^ (w + l) «"| = 0,
whence r = 0, and r = a\- 71 . (n -i- 1)}'^ .
If - be a fraction with an even denominator, it is obvious
2
that n . {n + 1) must be a negative number.
Let 71' + n = -p; .-. w + -I = y/i. _ p ;
.-. 71 = - ^ i \/:5: - i^ ; •■ p must never exceed 1.
If /> = |, ?2 = - i, and r = —=, or ^ = - the equation
Vd r
to the lituus.
MULTIPLE POINTS. 207
MULTIPLE POINTS.
202. Whenever two (ir more branches of a curve pass
through a point, it is called a multiple point ; and a double,
triple, or quadruple point, according as two, three, or four
branches pass through it.
If the bi-anches intersect, as in figure (l),
which represents a double point, there will
be at P two tangents, inclined at different
dy
angles to the axis, and thus — ^ will have
doc
two values corresponding to one of ir or y. ^ ^
Should however the branches pass
through P, as in fig. 2. and touch each 0\ P J^
other, and the contact be only of the ~'\
first order, there will be but one value
of —— ; but as there are two deflexions a nt
dx
dry
from the tangent, there will be two values of -~~^.
^ d,r
203. Problem. If 7/. =/(<«", y) = be the equation to
a curve, cleared of surds, and there be a point where two or
dy ^ ,
more branches intersect, — = - at that pomt.
do!
Differentiate the equation, the result will be of the form
,, dy ^r 1 ,^ du du
M . -^+ N = 0, where M = — and N = ■ — ■ .
dx dy dw
Then since two branches intersect, — will have two values,
dx
but M and N will be the same for both. Let a and /3 be the
dy
two values of -— ;
dx
.-. Ma + iV = 0, and M (i + N = ;
.'. M (a - ^) = 0, and a - /3 does not = 0;
dy No
.-. M = ; and .-. iV = 0, and -- = - — = - •
dx M
CoR. 1. Hence the value oi p or ——may be found, by
the same method as that by which the values of vanishing
fractions are determined.
208 MULTIPLE POINTS.
N
Thus, since p = - ijt = - when .v =%f/ and y = b; differ-
entiating numerator and denominator,
Mip + N., , _ dN dM
J/i + J/o/) dy dx
or il/a/)^ + 9,M^p + iVg = ;
a quadratic equation, from which two values of j) may be
found, and which, if possible, shew that the curve has a double
point.
CoR. 2. If, however, M^ = o, iVj = 0, and M^ = 0, when
X = a and y = b, differentiate numerator and denominator a
d^y
second time, and putting q = -— ^ , we shall have p of the
dx
form
Mjq + Nop^ + N^p + Ni
p =
M^q + Msp^ + Mip + M.
N,p' + N,p + N,
hen X = a and y = b;
M-iP + M^p + 71/5
whence we have a cubic equation of the form
p^ ->r ap^ + 6jt> + c = to determine p ;
and if there be three possible roots, there is a triple point.
This process must be continued, if the numerator and
denominator again vanish.
Ex. 1. Find the species of point at the origin of the curve,
ay~ - .r'' - bx^ = 0.
dy
Differentiating and putting p for -— ,
dx
2ayp — 3x^ ~ 2bx = 0;
3x^ + 2hx .^
.-. » = = - , II X = and y = 0,
^ 2ay
therefore there may be a multiple point.
Differentiating both numerator and denominator,
Gx + 2b 2 b
p — = , when ,7? = ;
2ap 2ap
.'. p^ = - , and p = ^ V - ;
a n
.MULTIPLE I'OLNTS. 209
.-. the origin is a double point, and the tangents cait the axis at
angles, tan"' \-, and tan"' (- \/-|.
This example will be useful in shewing another method by
which multiple points may be found. Thus, if there be a
surd quantity which disappears from the equation y =/(.»■)
by making ,v = a, but which is found in the equation
— = (h(w), then — will have two values, while y has but
dx ^ dx
one, and there is a double point. For resuming the last
equation, and solving it with respect to y,
X , dy v^,r + 6 x
\/a d.v /y/a 2y/a\/x + b
Make w = 0. Then y = 0, and ^ = ± \/ ~ , as before.
d.v a
Ex. 2. Find the point at the origin of the Lemniscata.
(,/ + yy = rt2 (x' - y~).
Here 2 ( f + py) . (^'^ + r) = (i~{x - py) '■<
., p = 1 ^ = - ,. if Of! and y = 0,
^ d^y +2y{.v^ + y)
a^ - 2 (x^ + y') -2x{2x + 2yp)
~ a^p + 2p{x^ + y') + 2y{x + 2yp)
= - — , if .r = and v = ;
a p
.'. p^ = I, and jj = i 1,
dv dy -
or -— = ± 1 ; or — = tan 45, and tan 135.
dx dx
Ex. .S. Find the same, when x^ - ayx^ + hy^ = 0.
Here 4^^^ - ax^p - 2ayx + Shy^p = 0;
2ayx - 4>x^ ,
. ^ _ __^^ = _, when X and y = 0,
^^ 3bjf-a,7^ O'
2apx + 2ay - 12x^ _0
6bpy - 2ax
14 P
210
MULTIPF.R POINTS.
2ap + 2nxq + Sap — S^r
Ghp^ + Qhyq - 2a
4>ap
p^ - a\ = 0;
.'. jt> = 0, and
/> = =t V I,
there is a triple point at the origin, and the axis of w is one of
the tangents.
The triple point at A is repre-
sented in the annexed figure; TAt
is the axis of x, AT^ and AT2 are
the tangents of the angles,
tan
"' \/ - , and tan"^ f - \/ 7
Ex. 4. Find the same, when y^ - Sa.vy + ,v^ = 0.
y"p — axp — ay + x"' = ;
ay — of-
p =
if 07 = and t/ = ;
p =
= ^"^P , if^ = 0;
!/■ - ax
ap - 2a?
2?//) — a 2yp — a
.'. 2yp^ - ap = ap,
or p (yp - a) = ;
a a
.'. « = 0, and « = - = - = CO .
The origin is therefore a double
point, and the two axes are the tangents.
The curve is represented above.
dy
204. If the branches touch, then -^ will have but one
dw
value, and yet at the same time be of the form - .
■^
For supposing the contact to be of the 7i"' order between
two branches of the curve; then the values of the differential
coefficients, as far as the (n + 1)"' coefficient, when .v = a, and
MULTIPLE POINTS. 211
y = b, will be the same for both branches ; but after the n"'
they will be different.
Let M — + A^ = be the equation after the first differ-
d.v
entiation, the original equation being previously freed from
surd quantities.
Then, repeating the differentiation (w) times, we have
M being the same as before, and iV, being the sum of the
differential coefficients below the (n + 1)*'', together with func-
tions of X or y.
But ^ has two values, as « and j3, while M and iV,
remain unchanged ; .-. M . (a - /3) = ; .-. M = 0.
^ ,^dy , dy
But M— + N=0; .-. A^=0; and .-. -^ = - .
dii7 ax
The analytical character of double points of this descrip-
dy d'y
- = - has but one value, — .
d.r d.V'
1 , ^V , , , d^y
tion IS, that when - = - has but one value, — .. , which also
,
- , has two.
CONJUGATE OR ISOLATED POINTS.
205. Conjugate or isolated points are those which have
a real existence, and are determined by the equation to the
curve ; but from which no branches extend.
Hence if cT = a and y = b give such a point, then x = a + h,
and .V = a - h, will make «, — , and the other differential
dcV
coefficients, impossible.
dy
■ +h) = V + -, - A +
rf ,r
d" y h"
■ + — ^ + &c.
dcV" l...n
is impossible, and that y and h are possible quantities, it is
evident that some one of the differential coefficients is im-
possible, when a? = a, and y — b.
d y d- y hr
Also •.• y, =f{.v +h)^y + £h + ^ ^ ^ + &c.
212 CONJUGATE POINTS.
Prop. At a conjugate point, if the equation be freed
oi surds, — = - .
dx
For differentiating the equation, u = f{wy) = 0, we have
j/^ + Ar=() (1),
d,v
d y . . d'^ y .
and if — ^ be not impossible, let be impossible ;
.•• continue the differentiation of (1) {n - l) times;
... J/^ + AT, =0.
dw
Let — ^ = a + /3 \/ - 1 ;
dx''
.'. Ma + N, + M(i a/"^ = ;
d V
.-. J/ = 0; .-. from (1) iV= 0; .-. — ^ = - ,
^ dx
and the values of — may in general be found, if any, by
dx
the method used for finding multiple points.
Ex. 1. ay^ — w^ + hx"- = ; .-. "iayp - Sw^ + 26.1? = ;
3x^ - 2hx .^
.'. p = = - > II cV = and .*. V =
2ay
6x-2b b
= = , if .1? = ;
2ap ap
h , /^
.', p'^ = ; and .". /J = \ .
a a
Now X = gives y = 0, while p = \/ . Also since
a
/,^ _ })
y = X y , if iT = ± A, the values of y are impossible,
and the origin is therefore a conjugate point. The same
result may be obtained by differentiating the equation
2/ = ,r V •
^ dy . /x -b a? / -b
For - ^ = V + 7- — ^^^~ = V - ; if cP = 0.
d.T? a, 2\/a\/x — b a
CONJUGATE POINTS. 213
206. The comparison of this example with Ex. l. in
multiple points will serve to explain the origin of conjugate
points. In the curve ay' - oo^ -hx~ =0\ two branches pass
through the origin and meet at a point a? = - 6, forming an
oval, while in the curve ay' - oo^ + ba>^= 0, the oval disappears,
and no curve exists between the values of a? = 0, and x = b;
these cases are represented in the annexed figures.
\
These two examples will shew that points of this kind
arise from the vanishing of certain portions of the curve,
owing to the change in the value of the constants.
Ex. 2. y - b = (w - of \/cV - c ; aa<2a, i/ is negative,
X — 2a, y = 0,
a!>9,a, y is positive,
a; = oo, 2/ is 03.
Again, let - a; be put for x ;
.'. y = - X
X + a
To draw the asymptote
2a
is always negati'
^
X \ ( 2a\ , a .
a 2 a-
X x'
2 a
= a? - o -
.-. y = X - a is the equation to the asymptote.
Take .-. AB ^ a ^ AD, and the line BD produced is th(
EXAMPLES. 219
asymptote. Also take AC = 2a. Then since y = 0, both when
cT? = and ,v = 2a, the curve cuts the axis at A and C.
Between A and B the curve is above the axis ; at B the
ordinate is infinite ; from B to C the curve is below, from C
to infinity it is above Ax. Again, since if x be negative, y is
negative; the branch on the left of A is entirely below the axis.
To find the values of -— ;
d.v
dy .V - 2a (.r - a) - (.v - 2a)
- = \- OC . ri;
d,v X - a (*• - ay
a^ - Saw + 2a- aoe 'V~ - 2aA' + 2a" ,
{X - ay "*" (v - ay " (x - ay
Let ,v = a; .-. — = co ; and the infinite ordinate through
dx
B is an asymptote; if j? = 0; -j- = -^ o^" angle at which the
dy
curve cuts the axis at A is tan"' (2); if x = 2a, — again = 2,
or /at which the curve cuts the axis at C is = Zat A.
Since if (i?^ - 2ax + 2a^) or {x — a)^ + a^ = 0, the values
of a? are impossible ; there is no maximum or minimum or-
dinate.
d^y 2.{x -af -2\ {x -ay + a ^ _ -2a^
Again, — — = ;; r^ — 7 rzi
** . dx~ (x - ay (x - af
.-. — - is + if .r < a, and is - if x > a.
' dx-
But xa<2a, y is - ; and x>2a,
y is + ; therefore from A to B, and from B to C the curve is
convex, and from C concave to the axis.
If J7 be -, — - = is +, but y is -; therefore the
dx- {X + ay
branch from A to the left hand is concave to the axis.
^ .r^ + 1 X + 1 ^,
Ex. 2. Let y = = Ci" - a? + I) ;
X — I X — 1
/x +1 / ,
= ± V . \/x- - cV + 1.
220
EXAMPLES.
If l, y is possible ±,
•r = CO, y is X ±;
therefore there are two infinite branches extending above and
below the axis of positive abscissas.
For X put
/ .V — 1
• ; .-. y = ^ \/ {x^ + 0? + 1), which
X + I
is impossible, if x be < 1 ; and =0, if ,^? = 1.
If cr>l, and increase to infinity, y is possible ± and
increases to infinity ; therefore there are two infinite branches
which meet the axis Ax^ in a point C, if AC = 1.
To find the asymptote :
//==*= V_f... Vi-i + l
I X X*
EXAMPLES. 221
+ — + &c.
2'^ / 1 ^ X
v(l + — + &c.)
1 2,v 2 a)-
1 + &c.
2.1?
= ± {1 +^ + &C.S-K1 -^ + ^^-)
= ± .r h + — 4- — + &c. J ;
, 2 a! a;~
.'. y = ± {.V + J) gives the two asymptotes.
Take AD = AD^ = ^ and Ac = 1. Join cZ> and cZ)i, these
lines are the asymptotes, and if through B an infinite ordinate
be drawn, two branches of the curve will lie within the angular
spaces formed by the intersections of this line with cD and cD,
produced. For these branches of the curve will always lie
above the asymptotes, since the ordinate of the asymptote is
always less than the ordinate of the curve. This may be thus
shewn.
Let 2/i be the ordinate of the asymptote ;
.-. y" = -, and y^
x - 1
a?^ + 1 - (a? - 1) . (a? + ^^^ + j)
1
3.r + 1
y-y^-
or (.y + 2/.)(!/-2/0 4,^^_i)
But 2/ + 2/i is + ; ••• 2/ - 2/i is + ; or y>yi.
Similarly it may be shewn that the branches which extend
from C, above and below the axis C-^ ^'^'
„ . dy dy, d'y ^)^
dy d^y
where » = — and a = -r-^.
dx dx
This expression has two signs ; but if we call the radius
positive, when the curve is concave to the axis, or when q
is negative ; and if, on the contrary, when the curve is con-
vex, or when q is positive, the radius be reckoned negative,
(l + p^)i
we shall always have R = ^-^ .
The co-ordinates a and /3 may be found from the equations
dy'
dx^
and .r - a = - (« - p) . — - = . p ;
^^ ^' dx q ^
and the circle is thus completely determined.
230
THE EVOLUTE.
218. In the annexed figure, let
AP be the given curve, PO the
radius of curvature, and O there-
fore the centre of the osculating
circle.
Also let AN = .V ; NP = y.
Then An = a, nO = - (i;
(1 + F)
PM=y-(i =
OM = a - X
{i^jr)
PM and OM are respectively called the semi-chords perpen-
dicular and parallel to the axis of x ; for if we describe the
circle, of which the radius is OP and centre O, PM is half the
chord of an arc, since OM is perpendicular to it, and OM is
equal to half the chord drawn from P parallel to AN.
219. The point changes its position with the change
in the place of P, and traces out a curve, which is called
the evolute of the original curve. Hence we may define the
evolute to be the locus of the centre of the circle of curvature,
and its co-ordinates are a and /3.
And since from y =fix); p and q may be found in terms
of y or w,
1 +p^ I -^ p^
and .-. from y - ^ =
?■■>
9 9
and y=f(x); y and ,v may be eliminated; therefore there
will arise an equation between a, /3 and constant quantities,
which will be that of the evolute.
220. Since
+ {y
y-^
dx
dy
.(.r-a);
but this is the equation to the normal of the original curve,
drawn from a point, of which the co-ordinates are .r, y, and
passing through a point whose co-ordinates are a and /3.
Hence the normal passes through the centre of the circle of
curvature, and therefore the radius is coincident with the
normal.
RADIUS OF CURVATURE. 231
221. The radius of curvature is a tangent to the cvolute.
dy
Resuming the equation {oc - a) + {y - (i) . — = 0,
Differentiate it, considering y, /3 and a as functions of x ;
dw ^^'' ^^'dw' dx' dw'dw
da
da d(i dy dy _ dx ^ da
dx dx' dx ' ' ' dx dfi d/3 '
a?
/. (x-a)-{y-(i).^=0; or (^ - y) = ^. (« - .r),
which is the equation to a tangent drawn to a point, (/3, a),
and passing through a point, (y, x).
But (j8 - y) = — (a - A') is identical with
da
(!,-/3)=-g(.-a).
or with the equation to the normal of the original curve.
Hence the normal to the curve, i.e. the radius of curva-
ture, is the tangent to the evolute.
222. To find the length (s) of the evolute.
Since R' = {x - af + (y - (if.
Differentiate, considering R, y, x, /3 as functions of a,
da da da [ da)
= -{,.-„. (,-/3).^}
=''^-<-)i-a (■>•
232 THE EVOLUTE.
A„a«»-=(.-»r.(..S)' <^>-
Divide (2) by (1);
da da^ da '
.*. R = s + c, c being some constant length.
Hence, if the equation to the curve be algebraical, R may
be found in finite terms, and the length of the evolute found ;
or the evolutes of algebraic curves are rectifiable.
Cor. Let 5j, s^,, be two arcs of the evolute, from its com-
mencement to the points where the radii are ^i and R.^ ;
.-. i?i - Sj = c ; R.2 - S2 = c ; .'. R.,- Ri = s.^ - s^:
let *2 - *i = « ; .'. a = R2 - R^ ;
or the difference between two radii of curvature equals the
length of the arc of the evolute intercepted by them.
223. From this property it is,
that the curve has derived the name
of evolute.
For if we take a string of con-
stant length, one end of which is
fastened at B, and the remainder is
made to coincide with the curve
COO^B, then if the string be un-
wrapped or evolved from COO^B, it
will describe the curve APP^.
COB is called the evolute, and
APPy involute.
From this construction it is obvious,
(1) That the arc OOy is equal to P,0, - PO.
(2) That O is the centre of a circle of which the radius
OP is the radius of curvature to the point P.
(3) That PO is a tangent to the evolute.
(4) That PO is a normal to the involute.
224. Another, geometrical, method of finding the radius
of curvature and the co-ordinates of the centre of the osculat-
ing circle is to assume that centre to be the limit of the inter-
sections of two consecutive normals.
THE EVOLUTE. 233
The truth of this assumption may be thus shewn :
(« - j3) = - — . {cV - a) is the equation to the normal,
dy
or Cr - a) + (y - /3) . — = 0.
Now at the point of intersection, a and /3 remain the same
, dy
for the two normals, while x, y and -— vary, since at a con-
ax
secutive point, x and y become x + dx and y + dy ; therefore
differentiating, considering a and /3 as constant.
The same equation has been before obtained to find the co-
ordinate /3 of the centre, and a is then known from
dy
.-„=-(s,-/3)-.
And R may be found from the equation
R' = ix-af+(y-(if.
225. Hence to find the radius of curvature in spirals.
AP the spiral, S the pole. PO a
normal, and the point of ultimate inter- Y
section of two consecutive normals. O is
the centre of the circle of curvature.
SP = r, PO = R^
SN±or\ PO = p,. 'S
SY=p, SO = r,^
Now S(f=SP' + P(f-2P0.PN,
orr,' = r'+R'-2R.p; for PN = SY.
Then since SO and OP remain constant, while
SP and SV vary, and since p=f(r);
^dp „ dr
... = r-i2-^; .'. R = r.—.
dr dp
If OM be drawn J_ to PS, or PS produced, then PM = ^
the chord of curvature through S,
Sr dr p dr
and PJ/ = POx --=r.-, .i- = /). — .
SP dp r dp
234 SPIRALS.
226. To find R in terms of r and B.
1 1 1 rfr
Si"'=e^ = - + ^.— , Art.086);
dp \ 1 2 dr^ 1 rfV
dp 1 If, di- d'r\
But 1=1 (.^ + ^
jo^ r' V dd'
1 1 / „ dr-\S ^ ^
p^ r^ \ de-J
therefore dividing (2) by (l),
rdr „ V "^ dey
R
dp „ dr'^ d^r
^ dO^ dO'
an expression for the radius of curvature, when r == f{0).
CoR. Hence also to find ^-— in terms of r and 6.
dp
dp 1 1 / , dr^ (^'^\ , ^
For V /.-3 = -T r' + 2.— - -r.-— ...(1),
dr p'^ r^\ de~ dOV
f~r'V "^ d&'l
therefore dividing (2) by (l),
.(2);
= i chord =
''-U^O
dp ^ ^'' , dr' (P r
^ r' + 2 . r .
227. Evolutes to spirals.
The point O will trace out the evolutc, and PO is always
a tangent to it, and SN is perpendicular to PO, wc must
therefore find the relation between SO and SN.
KVOLUTES. 235
Now r,' = r' + R' - '2Kp (l),
and p,= PV=V^''-p' (2),
dr
and p=-f(r) (3), and ^ =»"• -^ (*)'
between these equations p, r and R may be eliminated, and
the resulting equation will involve r, , p,, and constant
quantities, which will be the equation required.
Ex. Let the spiral be the equiangular.
rdr r
Here p = mr ; .-. i? = — = — >
^ dp m
p, = \/r^ - p^ = r\/l - m^ ;
.-. ri' = r + i2' - 2 i?p = f^ + ^ - 2r''
.-. pi = wrj,
or the evolute is a spiral similar to the original, and described
round the same pole S.
228. When two curves intersect, we have seen that the
distance between them, measured along the ordinate is, (when
X becomes x + h) expressed by the equation
/\ = A^h + A.J^ + A^lv' + AJi'' + AJ^ + &c.
If therefore we put (- A) for A, we shall have an expression
for the distance between them at a point where the abscissa is
X - h: let A 1 be this distance ;
.-. Ai = - A,h + A^h^ - A^h^ + A,h* - A^h' + &c.
Now since h may be taken so small that any one term shall
exceed the sum of all that follow it ; we observe First, that if
Ai = 0, A and Ai have the same sign, or that in a contact of
the first order, the curves touch, but do not intersect.
Thus the tangent does not cut the curve, unless A2 = 0, or
at a point of contrary flexure.
Secondly. Let both Ai = and Ao = 0, or the contact be
of the second order. Then
A = Ai/i'^ + Aih* + Ike.
Ai = - AJi' -{■ Aih* - &c.,
236 CONTACT OK CURVES.
which have different signs, and therefore if the osculating
curve be below the given curve at a point where the abscissa
is X + h, it will be above it at a point where x becomes cV - h.
Hence the circle of curvature both cuts and touches the curve.
There is however an exception to this, when the radius of
curvature is a maximum or minimum; for then (as we shall
see in the next article) A^ = 0, and the expressions for A and
Ai have the same sign.
For if the contact be of the third order,
A = A,h' + A,h' + &c. A, = A,h' - A.K" + &c. ;
that is, A and Ai have the same sign, and therefore the oscu-
lating curve does not cut the given curve.
Hence it appears that, when the contact is of an even order,
the osculating curve both touches and cuts the given curve, but
when the contact is of an odd order, it merely touches it.
229. Prop. When the radius of curvature is a maximum
or minimum, the contact is of the third order, or A^ = 0.
Let p=j^, 9=-^,, and r = -^.
Then R
dx dx dw^
-9
dR
But if i? be a maximum or minimum, = 0;
dx
.'.3\/l+p\p- f—r = 0, „. . _ 2.
q^ 1 + p
But 1 + p^ + (y — fi) q = 0, and if there be a contact of
the third order, we must differentiate this equation, and put
the co-ordinates of the curve for those of the circle ;
1 + »-
.*. 2pq + pq + (y - (i)r = 3pq r = ;
1 +/
The same result as before, and therefore when A-^, or the
difference between the third differential coefficient of y = /*(,?•),
and of R'^ = (v - a)^ + iy - /3)', ocpials 0, or when the contact
is of the third order, the radius of curvature is either a maxi-
mum or minimum.
EXAMPLES. 237
230. If q- = 0, while p is not infinite, /? = co ; this is
the case at a point of contrary flexure ; for then the curve
changes from convex to concave ; the circle of curvature be-
comes a straight line, namely the tangent, and before and
after the point the radius of curvature is measured in opposite
directions.
EXAMPLES.
(l) Find the radius of curvature and evolute of the
common parabola.
dy 2a d^y 2a dy 4a'^
i/-=4aa?, -p- = — , — -= 7-T-= ^»
doB y dw^ y~ dx y
4a^ 4!a^ + if 4a(a + .r)
1 + p- = 1 + -^ = -^ = „ .
y y .y
But R = ^^ "^^'^^ = {^a.{a + x)}i ^ 2(«+jr)i ^ _2^ ^^^
-<1 4a' y/a y/a'
^^ ^' -q 2/ 4a- ^ a'
VtV
x - a = - p .{y - p) = = -2(x 4- a);
y Oi
!a - 2a>
•. 3x = a — 2a, or
^<-^)
But •.• y" = 4aa? ; .*. (4a^/3)3 = — (a - 2a) ;
4 4
.'. /3* = . (a - 2a)^ = x^^ ; if a - 2a = .r,,
^ 27a ^ 27a
the equation to the semi-cubical parabola.
(2) In the Conic Sections, the radius of curvature oc
(normal)^.
. JV
Normal = N = y \/l + p^ ; .-. \/i + p^ = _ ;
238 EXAMPLES.
(1 + p')^- N'
R
-q -fq
Now if the vertex be the origin, and the axis the axis of a?,
•*• yq + p^ = n\ .'. y^q = ny^ - p^y-
= n (2m.r + nx^) — (m + nwy = - m- ;
(3) Find the radius of curvature of the ellipse.
y = - x/a"^ - .r?^ ; .-. p
ha
-7 =
+ p^ = 1 +
b-QS' a* - {a/^ - b-) x' a~ — eV'
a? (a^ - x') a' (a' - x')
ba
CoR. Let Ri be the radius at the vertex, and R.2 the
radius at the extremity of the minor axis ;
.-. R^ = ^^ 7 = - , and i?o = — = - ;
oa a ba o
therefore, the length of the evolute of the elliptic quadrant
= 2i2 - III = — = — 7~ ;
o a ab
4
and .'. the length of the whole evolute = — - . («' - //).
U R he a maximum or minimum, — — = 0;
dx
.: - 3 \/a^ - e-oo' . e',v = ;
.-. X = 0, and x'^ = — ; but a? = - or > a is impossible,
e^ e
^ d'R , /-^ -— , d y- —
and - = - Se-y/a- - e .r' + 3e^ x---\/ a- - ex-
dx' dx
= - 3e~a, if X =0;
therefore i? is a maximum, wlien x = 0, or y = i b.
EXA3IPLES.
239
Hence, at the extremities of the minor axis the circle of
curvature touches the ellipse.
(4) To find the equation to the evolute.
2/V.a^
^ 1 + />' (a' - e'x') Vd' -x" y {a" - e^^)
y-(^=-zj = ha = h^
" \b} {aef' " \bj (ae)!'
(a^-e-x~\dy /a^ - eV\ h\v
{a" - e\v')
_ 0? = cT -
* * I ~ I ~ 7 ' * ' 2 2 4 / x4 •
.x.^ / ^ (aa)i (6/3)i
But -o + Ti = 1 ; • • 4 + — :7 = 1 '
« o (ae)3 (ae)3
.-. (aa)i + (6)3)'^ = {aef = (a' - h'f.
(5) Radius of curvature and evolute of cycloid.
ivrp=i/ I; .-. /=-^— -— ^
I
\y I a y y a
Note— ^O/) in the figure should be a curve.
240 EXAMPLES.
To find the evolute.
^ dv \/2ay - y^ /
,,, _ a = - (y - /3) :v^ = - 2y f— ^ = - 2 x/say
ax
... a= x-\-^\/2ay-y'\
da 2 (a -2/) rfy , . 2 (a - y) 2a - y
.-. — = 1 + . =^ • 3- = ^ + = •■
dx y/'iay -y^ dx y y
dx dx y
1^ _ a/^-2/^ / 2a-(-/3)
•'• rf/3" ^ y ^ (-/3)
Take Am =Xy= - (i, and mO = a = yr-.
da dyi _ /sa - a?i y/^ax^ - x^
The equation to a cycloid, of which the vertex is A^ and
the diameter of the generating circle = 2 a.
(6) Find the chords of curvature drawn through the
centre and focus of an ellipse.
Since if CP = r, and CY, ± to tangent = p,
b'
.'. 2 logp = log a^b^ — log (a" + h^ - r^) ;
dp r
pdr a^ + b'^ - r' '
, , , , 2«dr 2 (a' + b' - r) SCD"
.'. chord through centre = - — = — = - .^ ,
dp r CP
dr 2(d' + b'-r') 2CD'
diameter = 2 r -— = = —^„ ■
dp p PF
(7) To find the chord through the focus.
Here if SP = r, SV=^p; p' =
2a - r
EXAMPLES. 241
2 log p = log // + log r - log (2 a - r)
2dp 1 1 2a
pdr r 2a -r r.(2a-r)
, , dr 2r(2a-r) 2SP.HP 2CD'
.-. chord = 2p -— = ^ = — = — — - .
dp a AC AC
(8) Find the form of the parabola y = a + bx + ex",
which has a contact of the second order, with a given curve at
a given point.
Make the given point the origin : then the equation to the
parabola becomes y = bx + cc^ ; and let y =f{x) be the equa-
tion to the given curve, from which find p and q.
But from above — = 6 + 2cx ; and -— = 2c ;
dx dx^
.'. p = b + 2cx ; and q = 2c.
But at the origin x = 0; .'. b = p, and c = - ;
qx^ q I ., 2p p~\ f
.'. y =px + — = -. [x^ + —X +^\ ;
^ ^ 2 2V q q'l 2q
P'\ 7 / PV
^^2-J=2'l^%-j-
The equation to a parabola, of which the axis is per-
pendicular to the axis of Xy the co-ordinates of the vertex
p^ p 2
, and ; and the latus rectum = - .
2q q q
Cor. The general equation to the second degree, or
^ + {ax + b)y + cx^ + ea?+/= 0,
containing five constants, may have a contact of the fourth
order, with a curve. And should there be a point at which
a^ - 4c = 0, the osculating curve is a parabola. Immediately
before and after this point, a^ must be greater or less than 4c;
and therefore the osculating parabola is intermediate between
an osculating ellipse and hyperbola.
16 R
242 EXAMPLES.
EXAMPLES.
(1) If y^ + x^ = ax - ay; an equation to the circle,
V2
(2) In the cubical parabola where a^y = x^ ;
ba X
and in the scmicubical parabola where ay^ = x^ ;
(3) The equation to the hyperbola being y'^ - — {x'^ - a^) ;
(e'x'-a')^ ^ , . , , ■
R = ^^ — ; and the equation to the evolute is
ab
(aa)i - (6/3)§ = (a^ + b')k
(4.) In the parabola the chord of curvature through the
focus = 4»S'P; and the length of the evolute
\/SA
(5) If yx = a^, R = - ^ ^-^ , and equation to evolute is
(« + i3)3-(a-/3)^=(4a)i
(6) The equation to the catenary is 2y= a\e^ + e " ) ;
shew that the radius of curvature is equal, but opposite, to
the normal. R = •
a
2\^2ar
(7) If r = a (1+ cos 9) ; the radius of curvature = ;
4r
and chord = — .
3
(8) In the spiral of Archimedes, the radius = the chord
a
of curvature, when r = —7- .
v/3
EXAMPLES. 243
(9) The evolute of the epicycloid, of which the equation
IS p- = e~ , — ' , IS another epicycloid,
^ - a^
. o r^ - a," a^
Vx = e . ~ ; and «i = — .
(10) Find the chords of curvature drawn through the
centre and focus of an hyperbola.
(10) If. = CP; chord = '(^^-«^^'^>
(2«) U r= SP: chord =
r
2r{2a + r)
/ rl ^
(11) If y \/ 1 + — — = c, be the equation to a curve
df ^
(the Tractrix) ; the equation to the evolute is — — =
da c
(the Catenary).
(12) •In the focal distance SP of a parabola, take SQ
= PN ; find the equation to the locus of Q, and the radius of
curvature.
G
SQ = r, SA = a, z ASQ = ; r = 2ff tan - .
2
(/ + 24«'r'^+ l6a^)t
R
32a^(4a2 + 3r^)
(13) Determine that point in a cubical parabola, where
the curvature is the greatest : and the point in the common
parabola where it is 1^*^^ of the greatest curvature.
(1) ,r = -T^, (2) .v = 3a.
V 45
(14) If a0 = 's/r'-«- - asec"' [- I, (the involute of
the circle), shew that p = \/r^ - a^ ; and find the equation to
the evolute.
(15) In the parabola if D be the point where the axis
intersects the directrix, and PN, QM be ordinates of cor-
responding points in the parabola and evolute, prove that
DM=3DN.
r2
244 EXAMPLES.
(16) If X be the angle which the normal makes with the
axis of 00 in the ellipse, then
R
(1 -e'^sin'^X)*
(17) In the curve (hypocycloid) of which the equation
is 2?3 4- ^§ = ai ; the equation to the evolute is
(a + /3)U(a-/3)'^ = 2al
and R^ be the radii of
conjugate diameters of an
(/2i + R,l) (ab)i = a' + b'
(18) Let R and R^ be the radii of curvature of the
extremities of two conjugate diameters of an ellipse, then
CHAPTER XV.
ENVELOPES TO CURVES. CAUSTICS.
231. When a curve touches a series of curves, all de-
scribed after a given law, the former is said to be an envelope
of the latter ; these latter are of a given form, and the pro-
blem is to find the touching curve or envelope.
For the better explanation of this application of the Differ-
ential Calculus, let us suppose that it was required to find the
equation to the curve, touching any number of equal circles,
whose centres are in a known curve.
Then if y and x be the co-ordinates of the touching curve,
a and /3 those of the centre of one of the circles
But j8 and a are the co-ordinates of the known curve ;
••• i3=0(«);
.-. {y-cp{a)Y-^{w-af^'>^ (l).
Now if we suppose a to I'eceive an indefinitely small
increment, the equation (1) will belong to an equal circle,
the centre of which is indefinitely near to that denoted by
equation (l) ; and the two circles will intersect at a point
of which the co-ordinates are ultimately w and y ; and simi-
larly proceeding with a third and other circles, we may con-
ceive the touching curve to be formed by the continual inter-
sections of these circles : and to determine its equation, which
must be independent of a, a must be eliminated between the
equations \y—(b (a)}^+ {x - a)^= r^, and the equation which
indicates that we have passed from the consideration of one
circle to the other, that is, the differential of the equation (1),
taken with respect to a-
Hence we may conclude, that if F = /(ti?ya) = represent
the equation of one of the given curves, the touching curve
may be found by eliminating a between the equations
, dV
F = 0, and -r- = 0.
da
. dV
232. That K = 0, and — = (), are simultaneous equa-
da
246 CURVES TOUCHING CURVES.
tions, may be also thus shewn. Resuming the equation to the
circle.
Let a + ^a, and /3 + ^/3 be the values of a and /3 in the
consecutive circle ;
.-. {^v - (a + ^a)Y +{y-{fi+ S(i)\'= r^ ;
therefore by subtraction,
(.^, _ af - {^ - (a + ^a)Y + (y - fif - {y - + Sfi)Y = 0,
or ^a{2Xv-a)-^a\ + Sfi {2 . (y - (i) - ^(i\ =0,
or 2 (.. - a) + 2 . (2/ - )3) ^ - { ^/3 1^ + ^a } = 0.
da ca
Now make ^a = 0, and Sfi = 0, in which case the point
of intersection of the two circles becomes a point in the touch-
ing curve, and -^p- becomes the differential coefficient of 3 with
da
d3 d(i
respect to a, or = -— , and we have 2{x-a)+2{y -3) — =0,
da da
which is the diflFerential coefficient of (.r - aY + {y - )3)" = t~
with respect to a, between which two equations a may be
eliminated.
Prob. I. Find the curve which shall touch all the straight
lines defined by the equation y = ax + r\/a^ + 1, r being a
perpendicular of constant length from the origin upon the lines.
Differentiating with respect to a ; x and y being constant^
ra r \/a~ + l
vVTi '*'
x/r'-x' y/r'~x' s/ r^ - ^
.'. y^ + x^ = »-\ the equation to a circle.
Prob. II. A given straight line slides between two rect-
angular axes, find the curve to which it is always a tangent.
Let c be the length of the line, a and /> the parts of the
axes cut ofl' in any given position of the line ;
EXAMPLES. 247
f + ?f = 1, and a' + b~ = t? :
a
•'-
A' y dh ^ db db a
or
b^ ^ .V \ .t?3 j
'^^ -be ^^
\/y^ + x^ ' \/oo^ + 2/^ '
0? y \/y^ + x^ , ., 2 -
.-. 0^3 + 2/3)1 = c, and .v^ + y^ = d.
Prob. III. If the equation of condition be a" + 6" = c%
then the equation to the touching curve will be
<-»"+' + 2/""*"* = c"+'.
Prob. IV. Find the curve which touches all the lines
defined hy y = mx + \/'m?(^+ bi^ ; a and b being constant.
a'f+b'x'=a'b\
Prob. V. Find the curve which touches all the ellipses
described round the same centre and with coincident axes,
the rectangle of the axes- being a constant area (m^).
x' y'
Here - + fi = l 0),
a b^
and ab = m^ (2);
.-. ^xy = m^, the equation to the rectangular hyperbola.
Prob. VI. Find the equation to the curve whose tangent
cuts off from the axes two lines the sum of which = c.
\/x + y/y = \/c.
Prob. VII. Find the curve which touches all the curves
included under the equation y = xtanB ^ , being
supposed variable.
./ = 4-h(h - y).
Prob. VIII. Find the curve when AD" = «'""' . AT.
248
EXARIPLES.
Prob. IX. Find the curve, when the rectangle con-
tained by two lines, drawn perpendicular to the axis of .r, one
from the origin, the other from a given point in the axis, to
meet the tangent, is = b^.
Prob. X. Find the curve whose tangent cuts off from
the axes a constant area; the axes being first rectangular,
secondly oblique.
Prob. XI. *Pmd the same as in problem 7, when /* and 9
both vary ; but m^ = h* sin^0 .cos9; m~ being a constant area.
^ 27
Prob. XII. Two diameters of a circle intersect at right
angles, find the locus of the intersections of the chords joining
the extremities of the diameters, while the diameters perform a
complete revolution. Ans.
.,- = -
CAUSTICS.
233. By the same method as that used in the preceding
article, the equations to the curves formed by the intersection
of rays reflected by a surface, or refracted through a medium
may be found. These curves are called Caustics. Some of
them may be practically exhibited by means of a ring of metal,
placed on a sheet of paper and held towards the rays of the
sun : the curved part of the sugar tongs may be used for the
same purpose.
234. Prob. Rays of light fall perpendicularly to the
axis of X, find the equation to one of the reflected rays, and
the equation to the curve of their intersection, or the Caustic.
CAUSTICS. 249
QP one of the incident rays,
Pq a reflected ray, making with the normal PG,
LqPG= ^GPN.
AN= X, = z NPG,
NP = y, Y and X the co-ordinates of Pq ;
.-. Y - y ='ni {X - a?) is the equation to Pq.
But m = tan PqoB = tan (90 + 20) = - cot 20
'l-p^\ , dy
•■'here p
m
2p J dx
2p
the equation to the reflected ray.
dp
dx
to the reflected ray.
Now differentiating, Y and X being constant, and q =
,....1
.-.K = ».■-'''
and from the equation to the curve AP, y=fix): p and q
dy ^^^ d^
dx dx'-
and then between (l), (2), and y = f(x), y and x may be eli-
minated, and the equation between Y and X which is the
required equation to the Caustic, may be found.
Cor. 1. If the incident rays proceed from J, the origin of
co-ordinates, we shall find by a similar method that the equa-
tion to the reflected ray is
■ •' 2/)j, + ,r(l -p')'-
250 CAUSTICS.
Cou. 2. To find the length Pp of the reflected ray.
Pp'^{X-wf+{Y-yf
.-. Pp = i.-JL^^ or, (Art. 218),
-q
the length of the reflected ray = ^^ the chord of curvature
perpendicular to the axis of co.
Hence, we may construct the caustic : take OP the radius
of curvature, dravir Pq = ^ the chord ± to .r, and making
with OP the same angle that NP does, then p will be a point
in the caustic.
Ex. 1. Let the rays fall perpendicularly to the axis of
the parabola.
g 2a 2a ^c^
y =^ax', p = -~;q=-—-p= -;
y y y
q 2a
„ 4a^ a CO — a
-/ = 1 - — =1--=
y OB X
-8a^ -8a^ -2a
2q
'•* '^aay xy
1 -p^ X - a iSa - x\
whence the caustic cuts the axis at the origin and at a dis-
tance 9 a from the origin.
Ex. 2. Reflecting curve a cycloid ; rays parallel to the
diameter of the generating circle.
AN=x., NP = y, BC = 2a.
n.u ^y V2ay- y"
Then » = -— = — ^
dx y
y~
X = a - ^ = .r + ^ \/2ay - f (I),
q a
CAUSTICS.
251
r =
•• y
dX
9.q
a - s/ (T - aY,
-(2ay-y-) (2);
a
(S).
From (1) -—=-y/2ay-y' = ~^^Y.
ay a v^a
From (-3)
dy a
''' dY
V a
\/a-Y
Y
the equation to a cycloid of which the radius of the generating
circle is a^ and therefore the base = AB.
Ex. 3. Let the rays fall parallel to the diameter of a
circle, find the caustic. The Epicycloid.
235. When the pole S of a spiral is the focus of incidence,
to find the length Pq of the reflected ray, and the Caustic.
252 CAUSTICS.
PO the radius of curvature = i?,
SP = r, z SPO = 0= Z OPq,
SY = p, Pq =p;
q being the point of intersection of two consecutive reflected
rays, and therefore a point in the caustic.
Now S, 0, q may be supposed to be fixed while P moves
through an indefinitely small arc.
Also S(f = / + /^^ - 2i2r cos 9 (i),
Oq^ ^p" +R^ -2Rpcose (2);
dr dr
r. r —- - R cos e —^ + Rr sin = 0,
dO dO
P%-R-osei^^^Rp.me = o,
.-. -— (r - /? cos 0) = - Rr sin B,
du
-^ (o - 2? cos 0) = - Rp sin 0.
do
But r + |0, for a very small variation of P is constant ;
= — -^ ; hence by division, we have
d9 dd
p R cos 6 — p
r r - R cos 9 '
Rr cos 9
rd through -S' =^ />
dr
" ^ 2r - Rcos9'
But /^ cos 9 = ^ chord through S ^.
dp
••• P =
dr
1^2 rf;,_l^.// yN
f) p dr r dr \ ^ r J
whence p the length of the reflected ray may be found.
CAUSTICS. 253
236. To find the equation to the locus of q.
Join Sq ; draw Sy J. to Pq.
Let Sq = 1\ ; Sy = p^;
.-. «i = rsni20 = 2r V 1 -^,- - = ^-^^ — ...(l)»
ri^ = r- + p^ - 2rp cos 2
= (r + p)'- - 4r|0 cos-0
^C^^rf'-^f" W;
whence from the given equation p =f(r) ; and from p = ^(r),
p and p may be found in terms of r; and from (1) and (2)
r may be eliminated, and the equation found between p^
and r,, which is the equation to the caustic.
Ex. 4. An indefinitely small reflector is placed in a cir-
cular ring. Every other point of the ring is luminous, find
the caustic.
Here p = — ; 2 « = diameter of ring ;
^ 2a ^
p~ r
r ia
p d
. log I — 1=3 log r — log (4a^) ;
I lf\ 3 r
2a
Hence the caustic will be a circle, the diameter — .
3
Ex. 5. Let the reflecting curve be the equiangular spiral.
p = mr ; — = TW^r ; log ( — ) = log r + log w^ ;
1 1
.-. - = -; p = r,
p r
Pi = 2m s/r^ - m?r~ = 2mr y/ \ — m^,
r^ = 4/ - 4jt>^ = 4r^ (l - m^) ;
or the spiral is a similar equiangular spiral.
Ex. 6. Let the radiating point be the extremity of a
diameter of a circle, find the caustic. The Cardioid.
254
THE CAUSTIC
237. When rays of light fall upon a plane refracting
surface, find the equation to the caustic.
QR an incident!
RS a refracted J ^^^ '
QA ± to BAC the surface ;
Q the origin of co-ordinates;
AQ = a; zRQA=e; ^RqA = ^;
.'. y=- .T tan + C is equation to Rq.
But .v = 0; y = AR = atane=C;
.-. y = - OS tan0 + aianO (l),
also sin = m sin (2),
since the sines of the angles of incidence and refraction are in
a constant ratio.
If now and (p be supposed to vary slightly, while y
and OB remain constant, . the intersection p, of two of the re-
fracted rays will be found, and p will be a point in the caustic.
From (1)
?rf0
,dG
COS" cos^ 9 '
(2) cosOdO = m cos (pd•= -, — ;
and similarly may other coefficients be found.
240. Take the expression for the radius of curvature.
^d^ -q
dx'
17 s
258 CHANGE OF THE
1 \^
h^
Ex. Let y' = 4wz = - a?,
d0 d6~
dy dy dO 1 dy
dos dd dx sin OdO^
d^y dy d~x d^y 1 dy
d'y dff'~'dx''de^ d?^ " sin0 ' ^°^ dO
d?^ d^ " sin^e '
1 d^y cosO dy cos 6 dy y _
■'■ ^^'de-~sm'9'de^^^'dd'^sm''e~ '
d^y
which is satisfied by making y = Acos{9 + B).
244. Find the radius of curvature, when the arc is the
independent variable.
/ dy~\^
^ V ^ d^") , ds' df
R = :; ' and -— , = 1 + -j-r. .
d-y dx' dx'
~dJ'
But if X and y be functions of s,
(dyV
d
dx^^'"-
(dxy \dx) \\dsj [dsl y
262 . CHANGE OF THE
do'V' idy
(:-:)"*(
di) ='
-(-sr=(£
and -
ax
dy d^oc do) d'y
d^y dc' ds^ ds' ds~
©■ '
1
dy
ds
d\v dx d^y
' ds' ds'ds-^'
.'. R =
whence by multiplying the numerator and denominator by ds'
ds^
R =
dyd'.v - dwd^y^
where dy, dx, d'y, and d^x are the first and second differ-
entials of y and x with respect to «.
OAK A ' 1 dy d-x dx d'y
245. Again, ♦.- — =— ^. . — ^.
R ds ds ds ds'
1 _dy' id'xV^ dy dx d\v d'y dx'^ id-y^
R' ds- '\ds' I ds' ds' ds^ ' ds^ "^ ds' '
/d^x\- jd-yy jdx d'x dy d'y]^
~ U?) "^ WJ ~\ds'~d? '^ts'Jp^j'
dx- dy- dx d'x dy d'v
ds~ ds- ds ds- ds ds-
R ^ [ds'l ^ UW
R =
^^WHSi
CoK. If (Px, and d-y be put for the second (h'ffcivntials
of X and y, and wc midtiplv the numerator and denominator
by ds'-.
INDEPENDENT VAHIAnLi:. 263
ds'
i? =
y/{d^.vr + {d'yr
Ex. Find the radius of curvature to the catenary.
TT /-2 ^ 1 (S + \/ii' + C''
Here iV = \/ c + s ; y = c log 1
\ c
dw s d~a' (?
ds ^/c^ + s'-'' ds^ (e- + *')S
dy c d-y - cs
ds x/c- + ? ds~ (jf + s^)^
d~7j\^ (d^i
di
c c
fj 7/ fJ 7/
246. Next, let u =fC%\ y)-, to find -— , and — in terms
dy dy
of r, and ^, when .v and y are functions of r and 0, so that
.^ = 0(r, (?); y = >i^{r,e).
du du dw du dy
dr dw dr dy dr
du du dw du dy
d0^ dw' dO d^'dd'
du dy du dy du (dw dy dy dw\
dr' dO dO' dr dw'Kdr dO dr dOl
du dw dti dw du /dw dy dy dw\
dr'de~J0'd^'^~Jy'\d^-'dd~'di''der
du dy du dy
du dr' dO dO' dr
dw
dw dy dy da
d^'dO'd^'db
du dw du
J^'de' dd'
1
dw
rf7
dw dy
dy
dw
(in
dy
dr' dO dr dQ
264 CHANGE OF THE
247. These values are much simplified, when
•T = r cos ; and y = rsin 9.
^
d'u dV dV , ^ dV cos9
•'• Zr~2 = T~ = "7~ • sin + — r . .
or dy dr d9 r
J. dV dru d-u cos 9 du cos 9
Hut -— = —— . sm 9 + .
dr dr- d9dr r d9' r^ '
INDEPENDENT VARIABLE. 265
d-u . ^ d-ii cos^9 d-u cofi'd du
.'. —-:= sin-0.-— - + — —'j^. + -T-
dy^ d'T r dO" r dr
2 sin 6 . cos 9
+
!d^u 1 du]
dddr ~ r'le]
7r
Similarly, or by changing into 0.
d^u „ d^u sin^0 d^u sin^0 du
—-5- = cos-0 . — — + — — . —^ + . -J-
dx dr'' r^ dO^ r dr
2 sin 0. cos ( d^u 1 du
r ' \d9dr ~r'd9\'
d^u d^u d~u 1 d^i^ 1 du
" dx^ ^ df ^ dr^ "^ r'' Jff' "^ r' dr ~
248. Transform the double integral fjVdxdy into one
where r and 9 are the variables, no and y being the same as
before.
•.• .r=rcos0; y = r&\n9;
.'. dx = cos 9 . dr - r sin 9 . d9,
dy = sin 9 . dr + r cos 9 . d9.
Now since in integrating, one of the quantities y or x is
supposed to vary, while the other is constant, let dx = ;
.-. = cos . dr - r sin . d9,
dy = sin 9 .dr + r cos 9 . d9 ;
.-. sin 9 .dy = dr ; eliminating d9 ;
.-. if dy = 0; dr = 0; .'. dx = - r sin 9 . d9;
dr
.-. dxdy = -rsin^.d^ X ^-— r = - rdrd9;
^ sin9
.'. jjVdxdy^ -jjV,rdrd9.
Thus if r = e^'+2'' ; jj^^^y'dxdy = - ffe''rdrd9.
266 CHANGE OF THi: INDEPENDExXT VARIABLE.
_ ^„ dhi d'U
Ex. 3. If — — + -— ; = 0, and .f -^ tf = r", transform to
dx'' dy
an equation in wliich i is the independent variable.
d-u 1 du
Ans. + -. — = 0.
dr r dr
d'u d-u d'u , ^ ., ,
iiX. 4. It -— ^ + --„ + — — , = 0; r = ar + y + ;jr,
dx- dy- dz-
. d'u 2 du
then --;, + - . = 0.
dr~ r dr
^ ^^ efw rf-w d-2^
Ex.5. If — - + + - — =0; and
da;'' dy- d z'
a; = r cos ; y = r sin . sin ; ^ = r sin ^ . cos ;
transform into a function of r, 0, ;
assume ^o = r sin ; and use Ex. 2 ;
d^(ru) 1 d^?« 1 d I . ^ d2i\
••• »• , / + ^^, • T-: + -^ — -.— . sin0. — =0.
dr' sm-9 d(p'' sm dO \ duJ
See Camb. Math. Journal, Vol. i, p. 121 ; and O'Brien's
Tracts.
Ex. 6. Transform fffVdwdydz, to a function of r, 9, cp,
fjJVd.vdydz = fffVr^dr sin . d0 . dip.
LAGRANGE'S THEOREM.
249. Let u-f{y), where y = z + .v(p (y), am\ z is in-
dependent of ,f ; required u or f(y) in terms of .t?.
By Maclaurin,
a;^ .r^ U„x''
u=U^+ C\x + U. + U-i + &c. + + &c.
" 1.2 2.3 1.2.3...W
where f/p, U , 6^.., &c. are the values of ?/, — , — ^ , &c. ;
dx d,v'
when X = 0.
First, if -*■ - 0, V = .r ; .-. rr„ =y(r).
LAGRANGE'S theorp:m. 267
du du dy . du du dy
Now -— = -—.-—, and -— = —.— .
dx dy dx dz dy dz
dy , dy . . , ^/ d(p{y)
But ^ = a; . (p\y) • ^ + (//)' ^^^^^^'^ <^^ = —of '
d y (,).-, ••-rf,-i_,^'^,)'
du du dy _ , ^ du dy
Makeci = 0; .-. f7, = <^ («) . -^ .
Next, let (,)_ = -; ••• ^^ = ^ ^
^. .^.. d^.. ^-(S) '•|^^^^^-
©1
dx^ dxdz dzdx d.
dzi
£/.=
dz
And so may U^ be found; but to find U„,
assume
dx"-'
dz"-'
let 50(y)S"-'-^ =
d7t„_i
= dz '
d"-^u d"-^u„_i
d"t6 d"u„_i
d''u
- = dz"-
d-^.l'''"-)
„_i V dx J
dx"' dx.dz"
'dx dz"-'
LAGRANGE'S THEOREM.
.'. U„
.-..{i,(.>,.i-/i.)}
dz"
Hence if the assumption be true for w - 1, it is true
for n ; and it is true for w = 1 and n = 2 ; therefore it is uni-
versally true, and writing Z for — , we have
dz
^ d\\[_cpiz)J.Z\ _^_^^^ ^ d'-K\[cp(z)]^Z\ ^"
dz^ '2.3 ' dz"-^ 'l.2.3.n
+ &C (1),
which is the theorem required.
Cor. ^^ f{y)=y or y be required, then
df(,z)
f{z) = z, and Z
dz
^•niMl.^.uc (.).
^ dz' 2.3 ^ ^
Ex. 1. y^ - ay + b = ; find y or the root of the cubic
equation.
Here y = -+-.?/', and taking series (2),
••• ^=^ "'"^ *«=»';
..^(.-)=«r\ li>(z)\'=A {0{^)F=^', \'P(^)V = ^",
d^r dz" '
'\\^ = 10. 11. 12;j:';
EXAMPLES.
2()9
^ I 1.2 2 . .S
10.11.12 ,^ ,
H .2f.tr + ozc.
2.3.4
b , ¥ b' U f>' o 1
a ^ a a' a a
Ex. 2. In the same example, find y".
Here Z = w2f''-', (p(z) = z^, and using series (l) ;
.
1.2.3
Ex. 3. Find log y, when 1 - ?/ + a^' = 0.
t/ = 1 + a^, and u = log 2/ ;
. ^ = 1, .v=l, (p{y)=a'^, f{z) = log z; .: Z = -;
d\cb(z)V.Z , . 1 «'' n ^ 2 2
. — ^^^ ^^ =2A.a~\ = 2Aa^-a^\
" dz z z'
270 EXAMPLES.
= gA'a^ - 6Aa^ + 2d^ ; z = 1,
= a\9A^ -6A + 2);
••• l"gy = a + (2 J - 1) ^ + (9^' - 6A + 2) . — ^— + &c.
Ex. 4. Let y = m + esin y^ find y.
Here z = m, x = e, (p(y) = sin ?/, and f(z) = z, Z = 1 ;
^'. y = z + (p{z)-+ 'v.' '^ .-— + '^: '^ — + &c.
'^ 1 dz 1.2 f/;jr^ 2.3
(p (z) = smz = sin /w if .r = ; .-. {0 (^) p = sin"'^.? :
d\(p(z)]"
.'. — — ~- = 2 sin ^ , cos -?? = sm 2;^ = sin 2m if ,v = 0,
dz
{0(^)}^=sin^;.; .-. '^^^^^S.m'zcosz,
— ^"V-;; — — = 6 Sin ^ cos-;^ - 3 sin^ijr
dz^
= 6 sin sr - 9 sin^ijf = | (3 sin 3z - sin z) ;
e . e' e'
.'. y = m + smm .- + sin 2 m . - + | (3 sin 3 m - sin rn) —
+ &c.
Ex. 5. Let ,F. = ay + hf + of + e?/ + &c., find y in
term of x-^.
Here y = — - - . {if + -y^+ -y* + &c.) ;
.'. z = ~; .v= - -; cp (y) = y' + 7^+7 ?/ + &c. ;
Xi h ^ of)^_fic ., 5ly^ - 5ahc + d-e ,
••• y = r, .1',- + — 5 a;,-' = .z','' + &c.,
a «■' ffl^ Yt' '
a general formula for the inversion of scries.
EXAMPLES. 271
(In du /r d^u h^
Ex. 6. Let u + — h+ — - + - .. h &c. = ;
d,v d,v' 1 . 2 d.v 2 ..'>
find h in terms of u, and its differential coefficients.
du d~u d^u
Put p, ({, r, &c. for ,- , -—, , -—3 , &c. ;
rf.-p rf.T diV
u I /(//a^ r/i"* \
p p\2 2.3 J
u 1 , qh~ rh^
.-. Z = ; .P = ; (h(y) = + + tec. ;
p p ^ 2 2.3
[u q w 3q~ - pr u^ ^ ]
\p p' 2 p" 2.3 J
If a be a root of an equation u = \ and x an approxi-
mate value of a, so tliat x + h = a:, tlie preceding series may
be used to find a near value of the root ; and it has been thus
used by Lagrange. Thus if u = w^ - 2x^ + AtX - 8,
I u 3 or - 1
h =
• . (c/ - .V + I) 2*. (w^ - a? +iy 2
21.1'^ - 12a?^ - 6.V -t 3 U^
+ &c
■I'
2'\{.v^- ,v +iy 2.3
23
.'. u = ; a = ,1
16
and if 1.6"! be put for .r, a more correct value may be obtained.
3 23 , . ,
whence if d- = - ; .-. u = ; a = ,v + h = l.Ol nearly
2 16 -^
EN!) OF THE DIFFERENTIAL CALCULUS.
THE
INTEGRAL CALCULUS.
CHAPTER I.
1. The Integral Calculus is the inverse of the Differ-
ential, its object being to discover the original function from
a given relation between the differential coefficients and func-
tions of w and u. At present we shall only consider the case
in which the first differential coefficient — is an explicit func-
dx
tion of ,??, as (p\x), and u = cp(oo) is required.
2. The process by which u is found from — is called
dcV
integration, and when to be performed is expressed by pre-
fixing the symbol f^^
Thus if -^ = (.x), u = J,. (p(x) + C.
Also since if — = (b(a;) ;
.'. du = (p (x) . dw,
u is found by prefixing the symbol /, thus u = f(p{x) . d.v + C,
or since / is the initial letter of summa, the integral has been
said to be the sum of the differentials of the function.
Hence f(p (j?) . da?, and j^fpi^) mean the same thing; also
since jdii = w, we see that / and d indicate inverse opera-
tions.
The letter C, representing a constant quantity, is added,
since constant quantities connected with the original function
by the sign ± disappear in differentiation : and therefore,
wlien we return to the original value u, an arbitrary quan-
tity as C must be added, the value of which will be determined
by the nature of the Problem.
, . , f^f^
3. The simplest case is wlien -— = ax"'.
^ d.v
INTEGRAL CALCULUS.
Let u = A.v" + C; .: -^' =
(Lv
Wyi.r"-'= a.r'";
.-. a =n A, and 7u = n - l ;
.-. w = ??? + 1 ;
a a .
ind A = - = — ; .a.r'" = -
" ...- + C;
27a
w rw + 1 m + 1
or to integrate a monomial, add unify to the index ^ divide by
the index so increased, and add a constant,
.„du a a \ ^
Cor. 1. Thus also if — =aa?""'= — , u = . r+^'
dx x'" m — l x"' '
which is derived from the preceding by writing - m for m.
Cor. 2. The general formula fails when m = - 1, for then
a . x^~^ _ a
u =
+ C = - + C.
1-1
du a 1
But ifm=-l, — = = a .- .
dx X X
1 d. log X a d . (log x)
Now - = — ; .-. - = a . ; ;
.V dx X dx
and .-. a . j - = a . \ogx + C ;
however, the true value of — - may be derived from the
dx
general expression, if C be first determined.
For, suppose u = when x = h ;
ab"'^' ^ ^ ab"' + '
+ C, or C
m + I m + 1
^m + l _ J« + l _ , , ^
.-. u = a. , a fraction of the form -,
m + 1
when m = — 1, and of which the real value is
a log - = a log X - n log b = a log x + C \
the same value as that which has been just obtained.
4. Since if //, = log \f{''v)\ — log (^), where s; = f(^x),
dz dz
du dx dx , ^ ^
T- = — ; ••• / — ^log(^) + C.
dx z ^ T z
18 T
274 INTEGRAL CALCULUS.
Hence, if there be a fractional expression, in which the
numerator is the differential coefficient of the denominator,
the integral is the logarithm of the denominator.
Ex. 1. Let
dA- 1 + A- -" 1 +a>^
.'. u =^. log (1 + or) = log \/i + x^.
^ ^ du 2 a; - 1 , , o
Ex. 2. Let -— = ~ ; /. ii = log (a" - a; + l).
5. Again, since
dp dq dr ^ d ^
Adp do dr „) r d
J:,[dw doc dx ) 'J^dw
= p + 9 + r + &c.
or the integral of the sum of any number of differential
coefficients = sum of the integrals of each differential coefficient
taken separately.
du
Ex. 3. Let -- = J.r'" + Sci?" + C.r? + &c. ;
dx
.'. u = Af^x'" + Bf^x" + Cf.xP + &c.
J ^, 5 „^, C
= x'"+^+ »"+' + xP+^ + &c.
m + 1 n + 1 p +1
6. If -— = «"*. -— , where « is a function of x, find u.
da? dx
^ du ^ ^ dz
Since if w = «'"+^ + C, — = (m + n^sr™. -— ;
dA' do?
. dz %"'+'
.'. /^i?'» . — - = + C ;
dx m + 1
or to integrate a function of this description, increase the
index by unity, divide by the index so increased, and by the
differential coefficient of the quantity under the index.
EXAfllPLES. 275
EXAMPLES OF SIMPLE INTEGRATION.
, ^ ^ du .J ax'
(1) Let -— = ax ; /. u = .
^ ^ dx 4
, ^ ^ du a ^ ax~^
(2) Let — = —6 — (ix~" ; .•. u =
^ ^ dx x^ - 1
du ^ n -
(5) Let -— = a.i?" ; .. u = . ax
dx m + n
(4.) Let — - = (a.i?" + 6)'" a?""'.
dx
Let z= ax"
-\-h
; .-.
dz
Tx"
wax - 5
J.f 00^ +1 S J^x + I 3 J,. x~ - ,v + 1 '
m • dzi 4.T/ - 5 4.1? - 5
1 o integrate — = = ^— — ^— .
da; x^ - X +\ I 1\- 3
.2?-- + -
2/ 4
T 1 c/?« du
J-t't a?-- = ^; .•,-—=; and 4ci? - 5 = 4;jf - 3 ;
r4;^-3 ;^ 1
4 4 4
= 2 log (;jf"^+ - 1 - 2 x/stan-'— J=;
\ */ x/3
.^ — 3 ^ ■, , . 2 , , „ 2 2 ti' — 1
tan" '
rx — \5 ^ 2 2
/ -^^ = log (a; + 1 ) + - log (a;''^ - a; + l)
a/3 x/3
Ex. 4. Let
, /x/.-r- - .t? + i\ ^ 2 , 2,1? - 1
log I I _ —^ tan-i ;=-
^ V .1? + 1 y ^3 \/3
d?^ 1
rf.i? (.1? + 1) (x + 2)- (,i?"^ + I) *
T f7 ^ 5 C i/.r + iV
L,ct — = + 1 -I .
V X + I (x + 2f X + 2 X' + 1
1 =J.(x + 2)' (x~ + I) + \B + C. (x + 2)\ {X- + 1) (x + 1)
+ (Mx + iV) .(cr+ 1) (x + 2)~,
.1?= - 2; .-. 1 = 5,5. (1 - 2) = - 5B, i.e. B = ,
a;= - l; .-. I = A .2 = 2A; .. A = - .
2
2 5
+ (M.t' + A^). (x + \){x + 2)',
284 INTEGRAL CALCULUS.
(5x* + 18a?^ + 23a?2 + 18a? + 8)
(x + 2).{x + 1)
10
{ C . (c??- + 1) + (il/a? + N) (v + 2)}.
Divide both sides by (x + 2) . (x + 1), or a;- + 3,r + 2 ;
.-. ~ = C (x' + 1) + (.¥a' + N){x+ 2).
Q Q
Letcr=-2; .-. ^- = 5C; .-. C = - — ;
5 25
9(.i?'+l) 5x'+3.v + 4> (7x^ + 15x+2) ,,,
25 10 50 V T yv T ;>
(7^1? + 1) . Cv + 2)
or - ^ ^ \^ J = (j/^j, + iV) (a? + 2);
50
.f7 1
" J^V ~ 2 J, a; + 1 5 /, (a? + 2)2 25J^w + 2~ lb 'J^aF+\
= - log (.r + 1) + loff (x + 2)
2 " ^ ^ 5 cF + 2 25 ^ ^ ^
7 1 /~2 1 ,
log v a? + 1 tan " ^a;.
50 *= 50
13. If there be m quadratic factors, each = (x — a)"'+ /3^
assume
U ^ Mx + N M,x + Ni P
.-. U = {Mx + N+ (M,x + N) [{x - af + /3'] + kc.\
Q + P\(x-ay+ fi'\"';
and after determining (Mx + N), by putting (w - af + ^-= 0,
and subtracting (Mx + N) . Q from U; divide both sides by
the factor (x — a)' + (i\ and then proceed in a similar manner
to find Ml and iV,.
1^ T ^ 1 , . . .
l^x. 5. l^et —- = — , resolve it into its par-
V (.r + 1)^.1' + 1) ^
tial fractions.
RATIONAL FRACTIONS. 285
U _ M.V + N M,x + N, __P
V ~ (^^Ti> ^ oj-^ + i "^ ^1 '
.-. U = \ = {{Mx + N) + {M,x + N,){x' + \)\{x ^l) + P{x^ + \)\
Let ar = "x/ - 1 ;
.-. 1 = {M\/'^+ N) . (\/^l + 1) = - J/ + J/v/- 1 + iV\/^+ A^;
.-. iV-i1/=l, andA^ + J/=0; .-.7^=1=-^/; .-. J/=-l,
1+i (a;+l)(a?-l)=^^=(il/,a;+iVr,)-(^'+l)-(«+0+^(«'+0';
2
.-. J = (ii/,.^7 + iVi) Gi? + 1) + P (a^- + 1).
Let w = \/^ ; .'. + 4 = (Ml y^ + N,) (\/^ + 1)
= - Jfi + J/, a/^ + iS^i v^^l + N, ;
.-. iV,-il/i= + l, andiVi+JI/i = 0; .-. iVi=i, and iJfi=-iVi=-i,
.r=-l; ...i=Px2; .•.P = i;
f/ - a? - 1 , w - \ , x
= -1- 4- + J-
X — 1
14. To integrate the fraction -—z -, , divide it into
* (.2?^+ 1)"
X — 1
two others, -„ , and -. . From the former we
(.2?'+ 1)^ {x" + xy
have,
but f 5 which is a particular case of j-r^ — — must
be deduced from that integral, which has not yet been found.
du 1
15. Now to integrate -— = -—o ^ ' ^e may assume
^ dx (x^ + 1)"
/, (.r'^ + 1 )" (x' + ly-' 7^ (cv- + 1 )" - ' '
1 A(x^ + 1) - (2w - 1)^-1?' B
(x' + ly (x' + 1)» ix^' + 1)'
A + .g + {ig- (2^ - 3)A\x^
286 INTEGRAL CALCULUS,
.: A+B=U B-{2n-3).A = 0; A= -^ - , 5=—"^;
2n-2 2w-2
/•I 1 w 271 - 3 r 1
■'' J^(cf~+ 1)" " 2W -2 ' (x'+ 1)"-' "^ 2W-2 V^(cr2+ 1)"-' '
by this process jj-^ — is made to depend upon j-~ ~^,
and by substituting n - 1, n - 2, and n - {n - l) for w, it
will be reduced to -^ = tan~\T?.
Ex.6. Let 7i = 4, or let / — . be required.
r I _ 1 c^ 5 r 1
J,(a?^ + 1)' ~ 6 * (^"+ If "^ 6 V. (ZTTP '
/•I _ 1 -^ 3 r 1
J,{aP + iy " i ' (^'"+ 1)- "^ i v,(ZTi7 '
/- 1 _ 1 «^ 1 r 1
J,, {w^ + 1)^ ~ 2 * .'P=^+ 1 2 V^cT- + 1
1 cT" 1 ,
+ - tan X ;
2 /.,(l +.rO"-'
.T 2 w - 3 ," 1
~ (2w - 2) (5?=^ + 1)"-^ "*" 2w^^ /. (1 + xy-^ '
the formula of reduction which was obtained in Art. 15.
-, ^,^ • ^^ '^^'"
Ex. 3. T.O integrate
rfcT? (1 + x'^y
r x"" r ^_j X
L (TT^" "^. '^'" (1 + .r^y '
^ dx ^ ^ ' ' {2n - ^) {\ + xy-^
r x'" -a?"""' m-\ r a?"""^
" /, (1 + .r==)" " (2w -~2)(1 +.r=*)''-' '*' 27^ -2^0"+ -i?-)""' '
RATIONAL FKACTIONS.
a formula of reduction by which the original integral may be
made to depend upon
r '^ ^j. r i^_
^,,(1 + xy^ ^^ J (i+.tj^)". '
according as m is an odd or even integer.
_ ^ du oj^
Ex. 4. Integrate — = ;
" rfo? (1 + xy
r x^ — x^ 4 /- x^
''' A (1 + ar^y " S(l+xy "^ 8 'Jx (1 T^^)3 '
r x^ - x^ 2 r X
y^l + xy ^ 6{\ + a»f "*" 6 V (1 + (v'y
6(1 +xy 6 1 +a?'^'
7. (1 + ^y ~ ~ [sil+x'f "^ 6 . 8 (1 + ct;2)2 "^ 6.8.(1 + .^/
1 r .3?^ 2^7=^ 2)
~~ %{\+x')'\{\+xy "^ 3(r+^2) "^ 3/
4 + 6a?^ + 5 — -, or 4b^;
c 4<(f
•••« = -•/
C 'J z
c '' z „ ^ac-b^
z^ +
4 c'
r 1 1 z
But ••• / — — =-.tan-'-;
J zZ^ -^a^ a
a
1 2c / 2cz \
-. ■ ^.tan-^ -— ==
c V 4ac - b- \\/4ac - b^
a b" , o b^ - 4'ac
-<-— , make a' = — ^— — ;
c 4c^ 4^^
c *>'z^^-a 2ca ^ z\z - a z + aj
2ca °\z + aj
1 2ca? + 6 - '\/6' -4ac
" x/6' -4ac ' °^2c.^? + 6 + v/62-4ac'
Ex.1. Let a = 6 = = 1; .-. \/4ac - 6=" = \/3,
/- 1 2 ,2a? + 1
/ -= -^tan-' —7--.
Ex.2. Let c = 6 = 1, anda = -l; .'. x/fc^- 4ac = \/5 ;
/- 1 _ 1 /2a? + 1 - \/5\
" J^ x^+ X -\ ~ y/5 *' \2.r + 1 + s/l' '
20. To integrate -~
(a + 6.1? + ca^Y
RATIONAL FRACTIONS. 291
a?"'
y
e'
,v'
"
(a +
6a? + ex
(•^'
■ +
h
c
■^^"^)
b
= ^,
or
X
+ a =
^,
if
a
b
" ic'
.'. .r- + - a? + - = sr^ + -„ = {%^ i p~) ;
c c c 4 -^; then / -^ r^~ may be found by
the method used in Art. 15, or Art. 17,
a ^ r. { z - «)"' _ r {%- aY
must be integrated by the method of partial fractions.
21. Again, to integrate
a?"'(a + bx + ex)"
1 dx 1
Let X , ,
du 1 dw
* ' d ^ J?' " rf .f %^' (az^ + bx + ey
^m+2n-2
■'■ ^ " " i (a2^ + 6;!r + c)"'
which is a case of the preceding article.
^ . du 1 ^ du 1
22. To mtegrate — - = ; and -—
dx a^- 1 dx x"+ 1
Since when n is an even number,
a;"-! = (.x'-l)(a? + l)(.i?'-2crcos — +l){a^- 2.rcos — + 1).
n n
continued to the factor x^ - 9,x cos I - — — j tt + 1 ;
and when n is odd,
INTEGRAL CALCULUS.
4'
.r" - 1 = (.r - 1) (,t'^ - StTcos h 1) (.r* - S.rcos h l) ...
n n
continued to the factor x^ - 2, V - -, .sin tan 'f I
y,.i'"-l w *' wl /i \ . 2m7r /
\ sin /
n
Qmir. ^ / ., ^ 2m7r \
— cos log \/ X' -9.x cos — h 1 > •
The method is the same when ?^ is odd.
The same method applies to -— = ; and n odd,
w n 2m + 1
.??" + 1 = (a? + I) (x' - 2x cos TT + 1) ;
2m + 1
w even, a? + 1 = (.r^ — 2/-l.sin( ^-—^j 2\/-l.sin-—
/ — 2w.(r+l)7r . 27n7r . 2m(r+l)7r . 2m7r
=2\/ - 1 cos ^^ ^— . sin 2 sin . sin ;
n n n n
2 /2m.(r+l)7r\
.-. M =- . cos — ,
w V w /
2 2m. (r + l)7r . 2w7r
JV = .sin . sm
n n n
2 2m(r+l)7r 2m7r
. cos . cos
n n n
2 2mr .TT
— . cos ;
n n
2m . {)• + l)7r 2nii- .tt
a- . cos cos
AIx + iV 2 n n
2m7r n ^ 2rmr
or -2x.to% + 1 ,T?- -2,r .cos +1
n n
Cask 1. Let n be odd; .-. Q = x- 1, and P =^ J ;
also .V' =f{M.i + N).(,v- \) + J. --~^^ .
RATIONAL FRACTIONS. 295
^" - 1 1
Let 0? = 1 ; .'. — = w, and A = — ;
x - 1 n
- 2w.(r+l)7r Smr.TT
/ .rcos cos
\ r ^ ^ I n n
•^ .x^ — 1 n Jx — 1 n
.r~ —2 07 cos 1- 1
n
the latter integral is of the form / „
J ^oe - 2ax +
Moo + N
1
Ma+N
= Mlogv jc - 2aOB + 1 + tan" —
oc - a
VI - a^ Vv/l - a
but a = cos ; .-. \/l - a^ = sm .
n n
., Ma + N 2 . 2w(r + 1) . TT
Also — . .sin ^ 1 .
/- ^^' 1 r I 2 ( 2m(r+ Htt
/ ~. = ~ • / 7 X + - •{cos . ^ i—
/j,A'"-I W ^, (.37-1) W^ 7i
1 a/ 2 « ^^^'T ^ . 2m(r + l)7r
log 'V x^ —2x cos 1- 1 — sin -^
/ 2m7T^
I oc — cos \
,-. ^)
\ . 2m7r /
^ sm /
sm
n
where m must be taken from m = 1 to m = —
CoR. 1. If r = 0, we have /
>/,r.1?" - 1
1 , ^ ^2 2W7r, ^ / ., 2m7r
- . log (.t? - 1) + - . cos log \J ,v~ - 2x cos + 1
2rmr
■V — cos
2 , 2w7r , n
. sm . tan —
n n . 2m7r
sm — —
n
296 INTEGRAL CALCULUS.
24. If we add to the integral the constant quantity
. /2m
- sm (
n ) \ . 2w7r /
\ sin /
Since tan~' A - tan"^ B = tan"M -— ) ;
r ,v' 1 2 /2m.(r+l)7r\
/ = - loff (a? - 1 ) H — . cos I I
^.^a?"-! w n V n )
, / „ 2m7r 2 . 2m(r + Htt
log X/ '' - 2,x' cos + 1 . sm —
2m7r
w
25. And .-. / = - log (x — l) + - cos f
log \/ a?^ —
SwiTT 2 . 2m';
2^7 cos + 1 — .sin
n n n
tan X
2m7r
I - X cos
Ex.
r 1 1 1 ^ ^ 2 27r
/ —^ = - log; (a? - 1) + - cos —
J x' -1 5 ^^ ' 5 5
tan
27r
, / , 27r 2 . 27r
log V .t;2 - 2a? cos — + 1 - - sin —
5 5 5
2 47r . / 47r
+ - cos — log Y .v' - 2.VCOS h 1
27r
1 — .1' cos
RATIONAL FRACTIONS.
4.7r
297
. 47r _, I
fsin
- sin — tan
5 5
] 47r
1 - *■ COS
+ C.
26. If n be even, there will be two terms of the form
A B
and ; and A and B may be found by putting
+ 1 and - 1 for x in the equation
X' =f{Mx ^ N){cc - \)^A
ct?" - 1
and x' = f{Mx + iV^) (.t + l) + 5
x-\ '
a?"- 1
a? + 1
J = -, and 5=±-.
27. Also the function — , since the quadratic fac-
tors of the denominator are included in the general formula,
cr^-2crcos(~ JTr+l, may be integrated in a similar
manner, and will be found to depend upon the terms
n\ n
. r(w -r - l)(2rw+ l)7rl
+ sin V -^ ^->tan
%m + 1
•^ — 'iX cos TT + 1
n
/2m + 1 \
/2m + 1 \
- X cos TT
\ n J
+ C.
n
If w be even, m must be taken from r» = to w = - .
' 2
If n be odd m =
n - 1
there will be a simple factor (,r + 1), and a fraction ^
where A
(-1)'
298
INTEGRAL CALCULUS.
r 1
Cor. Hence / ^ will depend upon
2 . ((w- l).(2m + l).7rl , . / T,^ /2w + l\
-.s.n| ^ J.logV..^-2.cos(-^)
2 . {n - l).(2m+ 1) .TT
. sin tan"
»i n
. (2m + 1)
.r . sm TT
1 - .r cos
:^'-)j
TT + 1
+ c.
28. To integrate
.r" - 2.T'"cosa + 1
The quadratic factors of the denominator will be all com-
prised in the term x^ - 2a,' cos 6 + 1, where 9 = .
Let
M/v + N
+ -
y2n _ 2a?" . cos a +1 x^ - 2.r cos + 1 Q '
.-. .v' = {ar" - 2cc cos + 1)P + {Mx + N) . Q.
Let cos + v - 1 sin = 2r, and Q, be the value of Q ;
.-. z' = (Ms; + iV).Qi.
Now ,r^" - 2a?" cos a + 1 = (^-
\n n j sill
1 sin (r - w) _ 1 sin {n-r)Q
w sin n' sin
, j'(w - r) (2W7r + a)]
1 sin (w - r) 6^ _ 1 \ w j_
7Z sin w0 w sin a
and the integral is reduced to that of
1 rx . sin (r - w + 1) + sin {n - r) 9
n sin a J^ .r^ - 2 0? cos + 1
the form of which is known.
EXA3IPLES.
r \ X
(2) / = log . .
^^^ J,. \,x + a) iw + h)~ h-a "" \(w + ay]
X' «
(A,\ f J = ; : log (W + a)
^^ J,Gt7 + a)(.t- + 6)Cr + c) {h-a){c^a) ^'
b^ c~
' (a' + h) + —rr r log ('^ + '•)•
(«-6)(c-6) *"' {a-c){b-c)
300 INTEGRAL CALCULUS.
^^^ A o;^ + 6a;'' + 8a; 4 "^ .r (.r + 4)" '
J^ 0?' + 2.i?2 + a- ^ Vv + i) a; +1
r /F _ 3 p + 3\-
^ ^ ^^(0? + '2) (a? + Sf ~ ~ x^S "*" °^ U + 2/ *
r 0? 31 2/0? -2\
^^^ X(a,- - 2) (.17 + sy " ~ 5 /r + 3 "^ 25 °^ WTsI '
r x^ 5x + 12 /.v + 4y
^^^^ i, (^T2f(^T4? " ~ .r^ +6x+8 "^ ''^ VT^ j ■
r w^ + 3x + I 1 (a? - l)""
0?=^+ 1
+ 3
r 2a? , /cr"'
<^^> J^ (a?^l)(.r2)(.-^l) =^"g^^^^^>
1 f , , /^^^l , \
10 (. * ^ .r + 1 j
r 1 11 1 / 0? + 1 \
^'■^^ J^ .??* + 4ci7 + 3 "■ ~ 6^T1 "*" 9 °^ V^,^.2 _ 2.t? + 3/
^^^V
18's/2 \\/2/
3»^+ 0? - 2 1 1 5 1
r Sar + a? - 2 1 1
2 .r
3, Va?^+l
+ -log tan 'a?.
2 *» a,' - 1
(17)
EXAMPLES.' 301
^^^) i~v\\-^x)\{\^x + w') ~ \ + x'^ ""^ (1 + a^f
1 , /2a? + 1
—tan-' y^
1 2 - 2-- ----
^'^^ X .^«T .1?' - a?* - .r' ~ 4 (1 + .r) a?'
" /Tr^ (a? + \y 1 ^ _,
-«- loo- x/ ^^ —^ tan ^x.
/on^ / = H ;=tan ^x\/ - .
X"
tan-' I —
cr" ar 1
^^^ Xl+2a?2 8 8 8 *
r x^ / , 3a?\ 1 3
*^^ J^(l + a?')' V 4/(1 + a?')'
r 1 111,-,
/c/tA r = r + — 5 tan \v.
^ '' J,x\{\ + x^) 5X-' 3x^ X
r ^ __ f Jl _i-_i^^_L_
^~^^ IxHT^^'' \3x' 3x ^lll + x'
1 5 5x\ I 5 _,
+ - tan X.
2
X b
(c-^\ f = 2 . log v/a + hx + ex'
'*' \2c^ rl ' J^rO + bx + cx^
r__x .^4 2 J_tan-'(^^i^'l
^^^^ X(l + aj+/^)'" 3(l+a;+a^) 'Vs V \/3 /
f302 EXA3IPLES.
- ''^'- ^ - + —7= tan '
) v/3 2 - a;-
_ = — + - log , - + —7= tan
(31) /- i = — log +-tan-'-.
(,3) /'^4log(-^)^£^--i-tan-'^!.
J.-r-'^-l 6 ''(a' + l)y^>^+.i-+l 2v/3 1-2^2-1
1 ,1 , /'1^
- sec~^? = - .sec"' -
a a \a
31. Class II. Next, to integrate
1 1 1
^/a^ i a^ \/ w^ ± 2 ax .v y/x^ + a^ x \/ d^ - x~
0) If'" '
dx y/ar' + «2
Let y/x^ + a^ = x%:
2 log ,v = log ct^ - log (^^ - 1 ) ;
dx xz
dz z'^ - \
du du dx 1 dx
dz dx dz xz dz
z'- 1
andz.= -r^ = l.r|-^ L_Uilog^
1 , \/a" + x'^ + X , , (\/a^ + x~ + *)^
= ^-^og-7 r .. — =4-log ^
= log
(2) Since
Vo^ + X'' — X
y/a" + x^ + 0?
log {x + \/x^ + a-) + C.
1
V -r^ + 2 a,v \/(.r + of - ^ '
.-. f—=== = log (.r + a + y/x^+2ax).
•^ J V .t + 2 ff .?•
du
Let v/cf^ + a^ = «^ ; .-. x'^ = z'^ - a" :
IRRATIONAL QUANTITIES. 305
.'. 2 log X = log (z- - n^) ;
da.
xz
dz z'-a''
du du da; 1
dz dx dz ^-' - a^
J~% - a 2aJ,\z-a z + a) 2a z+a
1 , \/a)^^a^ - a I , '^'
= — .log .-. = — . log 7~ 7 ~ V 7-A
2 a « ^, V ^^'+a 2 « * (v//p- + a' + a)'
los"
(4) Hence, if— = --.^=._^, M = -.log -__^ ,
dx a)\/a'^-x^ fi Wa^-d+a'
32. Integ-rate — =
doo \/a + hx + ccb^
1 1
\/a ■\-bx + c.t' \/c ^ / a h
\/ - + - X +
c c
1 1
V 2c n 4.n-
2c/ c 4c-
1 1
. /7 M^ 4ac -A''
:hich being of the form
x/^ii" ± a^
•. M = — = . log i ( A- + — + V - + — + a.2 j )
>/c '^'U 2c c c }\/^ac-lf]
1 , /2cci? + 6 + 2 v'^c v/a + />cr + c./
log
33. Also
\/c V yj ^ac - }?
I 1 1
\/a + hx - cx' V^^*\./« ^•'p 2
c c
20 X
306 INTEGRAL CALCULUS.
c 4c' V 2c/ 4c' V
d. sin"' I- j
which is of the form
ft
... /- ^ ^
X
1 . , 2c
;= . sin
v/a + 6a? - ca?' \/c /4-ac + h^
4c^
, / 2ccr - 6
sm ~ '
^ C?M 1
34. Integrate -— = . .
dx iVy/a + hx + cw^
1 dx I
Let a? = - ,- .•.--= ;
/' 2Ccr - 6 \
\\/4ac + hV
d%
du du dx 1 1 - 1
dz dx' dz 1 / 6 c »^ v/aar' + fcsr + c
- V « + - + "i
= - r ^ ^ . (Art. 31.)
-^xs/az^ + bz + c
35. Integrate
dx x^\/a + b
X + ex
1 du du dx ss^ 1
Let X = -; .'. —— = = — ■ =^=^ X
a; dz dx dz ^as^ + bz + c «
6 b
az +
r % 1 / 2 2
•^zy/asi^ + bz + c « y. \/a^ + 6^ + c
/\ az -{■ - I
^\/asf' + 6» + c 2 y/az^ + 6^ + c)
= -- \/aa' + 6sr + c + — . f -—==
a 2 a J,y/az~ +
bz + c
the integral of whicli depends upon a preceding example.
36. Integrate
IRRATIONAL QLfANTITIES. 307
du 1
d'V (a + 6.r) \/c + ex
ae - be + bz~
; a +bx =
e
du e 2Z
T / z'- e
JLet z = y/ e + ex ; .\ w = — ; a+bx-
' ' dz (ae - be + hz^)z ' e , ( o ae - bc\ '
(1) Let ae> be ;
2 . / b , z \/b
.'. 7/ = - X V _ . tan-'-7=
b ae — be \/ ae -be
^ _,/ \/b / \
= -/= — , - tan — ^===\/c + e.r .
\/b\/ae-bc Ky/ae — bc J
(2) Let ae = u''\ Also n is assumed positive, for if not, let
1
X = —
u
IRRATIONAL FUNCTIONS. 311
du
Ex. 1. Let -- = .rVl +^-
ax
TT 1 m 4
Here m - 1 = 3, and w = 2 ; .'.— = - = 2.
w 2
Let I + X- = z~ ; .-. .r^ = sf^ _ j^ ^4 ^ (^2 _ ^y .
.-. x^^= {z^ - \)%\
d%
du , dx „ , „ ^ 4
.*. -— = x^z -1- = z^ (z- - 1)= z - z' ;
dz dz
5 3 5 3
_ ^ du 1
Ex. 2. Let
da^ x*\/l+x^'
Here — = , and - = - 1 ; .♦, — + i-
^i 2 q -^ n q
And —
x' \/l + x"" x^'s/x-^ + 1 's/a?-2 + 1 '
Let x~'^ -^ \ = z^ \ .'. x'"^ = —z. — , x~^ = ^ - 1
dx
du z^ — \ „dx
dz z dz
a?-^—= -(«2- 1);
•■»=-(?-^)=--G-')
\/l + a?^ fl + ' v /'
in which the integral depends upon one in which the index of
X without the parenthesis is less by n than in the original
function ; after s reductions we shall come to f^x"~"'~ ' . (a + bx"y,
*u ffi • . i? 1 • I. • a (m - sn)
the coerhcient oi which is - - - - • which va-
b(rn + m) - (s - \)n
nishcs when w is a multiple of n.
46. Next to diminish the index r ;
/^.r™- ' (a + bx")' = a[y ' (a + bx^'Y " ' + A/.cr'"-"- ' {a + bxy ' ;
but from (A), by putting m + n for m, and ;• - l for r.
IRRATIONAL FUNCTIONS. 313
•^ h {rn + m)
substituting this in the preceding equation, we have
X"'{a + b.V"y ma .mi, , „^r-l /T.X
u = — ^ ^ + L^ {n + b.v"y ' (B).
rn + m rn + m
By this formula successive unities may be taken from r,
and thus f^x'"~^ (a + bx"y may be made to depend upon
/r""""""* (a + 6a?")'""*; sn being the greatest multiple of n
contained in m - 1, and t the greatest whole number in r.
47. When m and r are negative the formulas (A) and (B)
will not apply, for the exponent of •=_!, n = 2, wi - ] = m ;
.■r'" ^ \/\ - ■v''^ m - 1
._.;
J../7
v/l - .XT'
314 INTEGRAL CALCULUS.
But in practice we proceed thus :
• / — * / „•» . . /J — it- 5 — .
— = (m - l)a?'""2; q = -\/l-ai^
... «^ = _ x'^'W^ - c2?2 + (m - 1) {.aT-'^'s/l - x^
= -.r'""'\/l -.27=^ + (w - 1) T- y g - (m- 1)m*;
.-. m / / = - .'»'""' VI - 00^ + (m - 1) / -7 — „ ;
/. \/l - a?' w^ m 7^ \/i - A-'-^ '
and by putting m - 2, wz - 4, &c. for m, the integral may be
reduced either to
or, to - vl -
2W + 1 2w + 1
1 2w - 2 „
Q2„-2+-r- ,A„-3,
2w - 1 " ■ 2w - 1
and A = [-7^-. = -\/l-x';
• ^"^^ ~ 12^ + 1 ^" {271 + 1) (271 - 1) "-" "
2w.(27l-2) ^ „ 1
. ^^ ^^ Qo„_, + &c.>
(2W + 1) (2W - 1) (2W - 3) " j
^^n^^ - 2) (2/2 -4). ..4. 2 ^^^ -^.^ ^ ^
(2W+1)(2W - 1)(2W -3)...5.3
316 INTEGRAL CALCULUS.
If Pin+i = when a? = 0, since then Qon = ;
2w. (2w-2)(2w - 4)...4. .2
• 0= ^^ — + C;
(2w + l)(2w- l)(2w-3)...5.3
whence by subtraction,
9.n . (2n - 2) (2w - 4). ..4. 2
"^2"+' " (2W+ 1)(2W- l)(2»z-3)...5.3
- y^-, «- ^ (2«+iH2«-i) '^-'^ "■=•} •
Let c^? = 1 ;
r ct'^" + ' from '*? = 0) 2w . (2W. - 2)...6 . 4 . 2
•*• j^^i""^ ' to ci?= if ~ (2w + 1)(2W- 1)...7.5.3
Cor. If w be infinite, we may make P.^^ = Pgn+n
TT 1 . 3 . 5.7, &c. _ 2.4.6.8, k c.
"'' 2 ■ 27176 .8, &c. ~ 3 . 5 . 7 . 9, &c. '
TT 2.2.4.4.6.6.8.8, &c.
or — =
2 1 . 3 . 3 . .5 . 5 . 7 . 7 . 9, &c.
which is Wallis's Theorem for the length of the circle.
du „ ,.^
50. Let -— = {a'-a!-y.
da;
... w = a' J {a' - w^y^ - jw^ . {a' - ^^) ^"
, , !L::^ .r (a" - w'f u
n + \ n + I J^.
by which ?* is reduced, n being odd to j {d~ - .v )" •
r - r «' r ^'^
Also / («'" - nry = / . - / 7/-^ ;;^
•:^V.,/+-..si„-.-.
IRRATIONAL FUNCTIONS. 317
J,r ' 4 2
If the integral be required between .r = and ,r = a
■nd' 1 TT d'
2
7/ ^ 4 7^ 2.4 2
J/ ^ 6 J ^2.4.62
1.3.5. l...{n - 2) .n 7rrt"+'
id / (a- - x'')~ =
51. Integrate
2 .4.6.-8...(w - l)(w + 1) 2
dw 1
dcv a;"'\/l +.%•*
r L== f ^=..
1 dq '^
Here /> = ^— , and -— = . „
cr +1 da? \/l + a?
^ —d g = V 1 + .'t
d. — ' "^" '^
\/l + a;^ , ,. rv/l + .a7'
r 1 Vl + .r-^ . ,. rv 1
.17'"+ ' /r *''" + '^ v/ 1 + cV' -^.r a?" v/ 1 + a-'
r 1 1 v^I + a?^ w r 1
For m + 2 put m ;
/- 1 1 s/x^cc^ m-2 r 1
'l^x"'\/l + a;-~ m-l' x'"-' m - 1* /^a?'"--\/T+^'
and therefore the integral may be reduced either to
C ., or f J- :, , according as m is odd or
even,
also / , , ' = log / J
318 INTEGRAL CACULUS.
f ^=.= f J = -^.0,-^ + 1
and
^x x^ y/ 1 + a-
du
52. Integrate
d'V ,2?'" v/,1?- - 1
1 ^ a? r 1 rf • V-a?'
do?
= TT- + (m + 1) . / ■ - (m + 1) . 7== ;
r 1 1 s/x^-l m r. \
therefore, writing m for (m + 2),
r 1 1 \/,t?^ - 1 W2 - 2 r 1
and therefore t< may be reduced, m odd, to
r 1 _i , ^ r 1 \/.t''-l
/ — ^ = sec .1", and w even, to / = .
Example. Find
ind f~~-^
\/x' - 1
/• 1 _ J s/x" - 1
l.spr,-i.
+ ^ . sec ' ,j? ;
/♦ 1 , s/x' - 1 3 s/x' - 1 1.3
••• / / = t • 1 + • s + sec"^i?.
-'^ a;^ \/x' - 1 ^ + - .p + - = sr^ + - - -— = ^' =t /3- ;
2 c c c c ^c^
1 r (^-«r .„ b
J^ \/a + ba,- + cx~ \/c J^ V z^ ± ^" '
which may be made to depend upon j / ■, ^ r.. -
56. Integrate
- = a,
c
dx ct-'" \/a + 6c» + c.r^
1 dw \ du ^
^ '^ ~ %" ' d% %^' dx z" \/ a%^ -\-b% -Y c"*
r ^
m-i
c
which may be integrated by the preceding method.
du 1
57. Integrate — =
d X \/c - X \/ 2 a X - x' '
1 1 V ~ c
\/c - X \/^ax - x^ \/c \/2 a x - x ''
1 1 , 1 .7? 1 . 3 X- ^ ,
= — — . — . J 1 + - . - + . - + &c. 't
\/c \/2aw-x~ '■ 2 c 2.4> c~
r ^^ .
and thus u depends upon / . = ; this integral is met
/rV2a.r-rbxf 2a{a+bx) a^2(a+bx)^
^2(a+b
r\/"'+bx \/a+bx b i\/a+bx-\/a\
(2) / ., = + — 7=log {"7=--T- 7"( •
J,^ x~ 'V 2\/a [\/a+bx+y/a.)
r x^ Ua+bxY 3 ]
(3) / -— — - \ a(a+bxy+a:{a+bx)-a^\
2 y/a + bx
(4) /-^^^ '_iog^;^*"-^"
^xoc y/a + bx y/a ^/a + bx + Va
, ^ r ^ 2 bx-a 2 . , /
<^) / /r— = -7-tan- V =-7=.sin-V
J^xv ox-a v a a v«
21
bx—a
bx
Y
322
INTEGRAL CALCULUS.
■s/* + 3x 3 's/ ^ + 3a? - 2
-7^ log
16 v/4 + 3\/l-.r^
'^ ^ J^^l^a;' \5 15 15j
/« ;»'' fa;^ 5w^ 5.v\ "
1 -a?'' + — . sin"'a\
10
(20)
r 1_ = _ I J 1 + _Li
J^ a?** */l + a;' \ 5 ^^'^ 1 •'5 'i'^ 1 5 a? J
v/i
*^, (a + hx^)i ~ [Saia + hw') Sa'\^a + 6^^
EXAMPLES.
3U
-V.3 1
(22) f- , ^ o^B = C^'' + 2) -7-=7-
(21) / - = - U' + - ; ^3-
(25) / = la)' + 4,r^ + — -, .
r .v^ fSaP 4.cf' .r\ 1
^^^^ J.rii'T^^ " Vio5 "^ 71 "^ ly (I + -t'o'^'
(28) T— -==- = log (2,1; + 1 + 2v/l + .r + .r^).
1 . _,/2^^-l\
1 1 /.r-3 + 2\ /2v/l -a; \
(^^) XoT:^K7n:r''72*''"V Tf^ /"
. 1 /2 + a; - 2\/l +« + .'P^
(31) / -^—^^^''^l V ~ •
^^'^ i.(TT^y7n^^'v/2' """v 1+^ /•
1 1 /Vl + d?2 + .»v/2\
(^^^ i, (T^T^K/r+Tr^^ " ^""^ ' '2\/r+.r-.r'^/ ■
Y 2
324 INTEGRAL CALCULUS.
r 1 /.r + 3-2\/l -x-x^\
/- .r 1.-1 (''^'\
r 1 ^-g
(42) J\/LU1 = sin-\r - \/l - x'.
*^.r 1 — iT
<*') ll ^r^ = 'in- - + log (, + Jrr^) ■
(44) f'f'V ^—^ = - sin-'.F - - (2 + .v) \/l - .■p2 „
(45) r-^=^= tan-^V— -3--
l^ 1 .1? + a
r 1 _ f 1 2 ^ >t? + «
/r(2a.r + .r^)i~ l3(2aci? + a?'0 3«'4aV2a.v + .r- '
(48) f ' - ^•(^-^+^)_
^,(1 +aj + ,r^)i^ 3v/i -^,r + .r^*
(49) V ^ J-^^^.^'-\-^SiL±Jl-
(50) /— -7^ = — + tan-'(\/.r').
(51) f'- = [- - 1 ) 2\/.v + 2 taii-'\/^.
^,. 1 + .V \3 J
EXAMPLES. 325
(.02) r J^=2v/.x-+- log —__=_-. -tan-'f-- [
r 1 \/2aa7-a,'"
^ , r *'^ / fScA' - sh\
(54) / , =rzr = \/ a-\-hx ^- cx'\ ~ >
-^x Va + hx+ ex- [ 4c- J
4ac - 36' faca; + b + 2^c\/a + b(V + cx^\
" -^^^ '"« I ^0 1 •
(.OS) Rationalize the integrals
/-cr^ + 200^ + .ra / \ r *'"
in (1) make x = ^^'^j and in (2) make (1 + w) = z'^.
(56) /-.Vii^-'tan-'V^ii^
^ ^ /, 1 - cP- 2 1 - .r^
--(2+cl-^)v/l -*■*.
4
/ L v/"^ = log VM(?^f
(57)
where y = \ — 7-,
r x'^\ _ 1 ^ _1_^ . -1 Aw «-2\
J,^'^ -1 v/l -«*•" + a?' a/^^''"'' " V <^'' - 1 /
, 1 1
make x = -
X z
CHAPTER IV.
INTEGRALS OF LOGARITHMIC AND EXPONENTIAL
FUNCTIONS.
59- These functions are of the form X (log a?)", X. a%
where X is a function of a'.
60. Integrate /^^. (log. x')".
Let/,^=P, f,P.- = Q, andfM--=B.
Then /,^ (log x)" = P (log .r)" - n . j,P . (log .v)"" '
rP I
and / - . (log coy-' = Q(log wf-' -(n-l).f^Q. (log .v)"-'.- ,
rQ 1
J - . i\ogxy-'=R {\og,vy-'-(n - 2) . f,R.(\ogxy-' . - ;
.-. f,x (log a^y = p (log xy -n.Q (log wy-'
+ n.(n - l).R. (loga')"-2- &c.
Ex. 1. /^■a?'"(log.t>)".
.v^-^' {logg er n
= . La?'" . (log cX')""' ,
and in this manner may the integral be reduced to
f^a;'"= ^ , if w be a whole number,
v'" + ' n
and .-. f,a;'\(\ogwy= |(log,z)" (log a')""'
n.(n-l) , w.(w-l).(w-2)...2.1
+ -} n;- (log a,)"-^ - &c. ? ± — ^^ ^-^ ^- .t?'" + '.
(m+1)- ' ^ ^ » (»» + !)'"+'
Every term of the integral vanishes botli when .v = and
.1' = 1, except the last, which vanishes only when ,v = 0;
LOGARITHMIC FUNCTIONS. 327
from A' = 0) 1 . 2. 3...(w - 1). w
.-. ;. (log.), to..= ir (m + ir-
r ^
61. Integrate \— , n a whole number
^,(log.r)»'
d . log X
X
dx ^. d.Xosx 1
Smce X -^— = A' - = 1
(log a;') " dx x
'd.{Xx)
- X .X 1 I dx
+
J:
(n - I) (log x)" ' w - 1 ' (log.x)""'
Let'^4^>=P;
dx
/, (log cX')" ~ (w- l)(logd7)"-' n-\J, (loga-)""' '
J^ (log .'r)"-i ~ (w - 2) (log a?)"-=* n -2' J^ (logcj?)"- '
where Q
dx
Xx Px
^ (log xy " (w - l)(logct;)"-' ~ (n- l){n - 2) . (log cr)""'
- &c.
(w -l){n - 2) (w - 3) (log xy
r ^y
in this manner the integral may be reduced to / ^
which cannot be integrated except by a series.
Ex. 2. Find [tt^—tt,-
J^(logcr)~
r X'" _ P"^'Tf ^ -^ _^ ^^ _^ r ^"'
/, (log .7?)- ~ -^^ (log xy ~ log .T? ^^"' ' J, log .T/ ■
Let log a; = ^ ; .-. a;' = e% and x"' = e"'~ ;
^ x"' re"" dx _ rf^ ._ r^^^~
J,i.\ogx J^ X 'dz J~ z J~ X
(r« + \yz' (m + If^^ , 1
328 INTEGRAL CALCULUS.
(m + ifz- (m + ifzs
= \0frz + (m+ 1) .Z + ~— + ; + &C.
^ ^ ^ 1 . 2^ 2.3^
(m+iy(\osxf (m + lf(]oewf
= log(loga.) + (m + l)log;r + ^ ~^+- ,3/ ^ +^c.
Cor. If m = 0, we have
r 1 , /, ^ , (log xY
/ 1 = log (log.r) + (log 0?) + , + &c.
>^^ log iV 1.2"
62. Integrate fj,a^ . X, X being a function of oe.
Since — ; — = a' Jog a = ^ a' ; .-. La' = — ;
dx A
,^ ^a^- 1 rdX ,
A A J dx
Let -r- = /^, -r- = Q, &c. ;
/.a'.P=^--./Qa-;
Ca'.^=a-,|---+--&c.j
Ex. 3. Let ^*"' . a" be required.
^ A
Ex.4. /■^ = -f .^^J^./-^;
^, a?" - ' ~ (w - 2)ci?"-" « - 2 ■ /^ a;"-''' '
Jxaf (w - 1) . cT?" - ' (w - 1) . (w - 2) . .r"--
LOGARITHMIC ttJNCTIONS. 329
Also
/
(w-l).(w-2).(w -3). a;"-'
A"-' r(f
"^ {n- i).(7i-2)...l V, x'
Ci A\v' A'x^ \
/ / 1 -r Ax + + — ^ + &C. I
1 A^x A^.v- \
- + A + + + &EC.
X' 1.2 2.3 /
A\v' A^x^ A'w'
= loff X + Ax + :, + — — r + - — 1 + &c.
^ 1.2^ 2.3-^ 2.3.4^
Ex. 5. Find / - . log (a + />.r).
log (a + 6i») = log a (1 + - a?) = log « + log (l + - x)
(b b^x' h'x' b'x' \
= log a + - a' n + — r tt + ^^- '■>
® \a 2d' Sa^ 4a^ /
/I
- log (a + bx) = log a- . log a
.,■ "^
/6 6^r^ b\v^ b'x' ^ \
+ _ a? h r— , + &C. .
U 2^a=^ 3-a^ 4.'x' I
Ex. 6. Find /^a'".
(wa'logcr)^^ (w^logcx-)^
a;"^ = 1 + wo? log X + ^-^ + J 2 3 + &t;. ;
.-. /^a;"^ = *• + w . /,.« log ^ + — . j,x' (log a;)-^
+ ^^.!.a^'{\ogxy^hc.
Hence, the integration depends upon j^a'"' (log.x-)'", Art. 60;
.-. /,cf log a; = '- . (log a; - i),
x^ { 2 . ll
ia'-' (log xy = 3 Ulog xf - f log X + —J ,
X (x log ..)"> = ^ ((log xf - I (log xy + "^,~ (log a;) - ^1
&c.
330 INTEGRAL CALCULUS.
and arranging the terms according to the powers of log ,v,
Lv"" = X + — s- + - ^ - &c.
(a^ nx^ n^x^ \ w^(loa; .
r loffti? . cos + (w - 1) . j^ sin"--0eos-(^ ;
and putting 1 - sin^O for cos-0
= -sin"-'0cos0+ (w - \).fgsiW'-~e - (n - l)/esin"0;
- . ^ sin"- '0. cos w-1 ^ . _
.♦. fg.sm"6= + Jgmi" ~6,
a formula by which ^sin"0 may be reduced to - cos0, or 0,
according as n is odd or even.
Suppose n to be even or = 2m, to find the value of
II (sin 0)-'" between = 0, and = - .
Let fe (sin 0)"" = P,, ; sin"^"'-' cos = a,„_, ;
1 2m - 1 _
2m 27W
CIRCULAR FUNCTIONS. ."J.S.'j
P,,
But
Q-.„-.
= both for = (
[) and =
2 '
.•• P-^n
2m -
2 m
-A.-2;
.-. P..,-. =
2m -
3
2 m -
2
•: P.m
(2 m -
2m
l).(2m-
.(2m -2)
■'^ P ■
hence
(2m -
-l).(2m-
- 3)...3 . 1
n
2m. (2m
- 2) . 4 . 2
1.3.
5. ..(2m -
1) ^
2.4.
6. ..2m
2
59.
du
Integrate — :
= cos"0.
Jq cos" = Jq cos""^ . cos
= + cos"-* sin + (n -\) . fe cos""^ sin~
= cos»-^ sin + (w - 1) /e cos""' - (n - \) fe cos"
cos"-' 0sin0 w - 1 .
= i- L cos""'' 0,
w n
a formula by which ^ cos" may be reduced to sin d or Q,
according as n is odd or even.
70. Let — = \— . Since sin^e + cos'^0 = l ;
* d (sin ey
r sin'' B + COS'' c^ _ r 1 r cos- y
■"• "^ " X (sin 0)" ~ " i (sin0)"-^' "^ J, (sin 0)" '
r cos^0 COS I r sin Q
7g(sin0)" " ~ (w - l)(sin0)"-' ~ ?i - 1 Jq (sin 0)"^ '
COS / 1 \ Y 1
•*• '^" "(w-l)(sin0)"-' ^ V^ "7r^J7,(sin0)"-'-'
COS B n - 2 r 1
(w- l)(sin0)"-' "^ w - 1 ' Jg (sin 0)"-=^ '
a formula by which w may be diminished.
71. If — = ; ;^r- , then, as in last article,
d 6 (cos 6y
sin^0
./e(cos0)"-- ^e(cos0)"
and
INTEGRAL CALCULUS.
r sin"^0 sin 1 /• cos
■I 6 (cos ey " (w - l).(cos0)"-' ~ w - ] VeCcose)"-' '
sin w - 2
II — 'Z. r
n - 1 Jg((.
(n - }) (cos0)"-' w - ] Jg (cos Oy-' '
72. Let -— = (sin GY (cos Oy, m and w both integers,
(sin ey (cos 0)» = (sin 6^ cos (cos 0)"-' ;
.. /.(si„e)»(cosey= (?''»^+^/.(siner"(cose)"
(sinm^+'CcoseV'-i w-1, ,
' ^ + f/,(sin0)"'(cos0)'-2-/,(sin0)'»(cos0)"
+ 1
n - J\ m. + ti (sin 0)"' + *(cos 0)""^
» ?/ = M = ^^— —
m + 1
+ l;^- fi^^^oy (cos ey ^
m +1 Jg
(sin^)'"^>(cos0)«-i n - l
.X(sin0)™(cosm"-^
m + n m +n
a formula by which the integral may be reduced to
j^(sin0)™, or ^ (sin 0)'" cos a
du sin'"0
73. Let — = ,
' de cos'0'
rsm'"-'9sm0 _ (sin0)'"-' m-1 r(sm6y"-''
Jg (cos0)" (n - 1) . (cos0)""* n -l ^^(cos^)""^
a formula by which the integral is reducible to a known form.
74. Let — = 0". sin 0.
* de
fg0' sin0 = -0" COS + n.jG"-' COS 0,
fg0"-'cos0= +0"-'sm0 - (n - l)X0"-~sin0,
/,0'-'sin e = - 0''-^cos 0+ (n -2) fg0'-'cos 0,
&c. = &c. &c.
X^'sin = - 0"cos + n0"-^sm + n (n - l)0"--cos
-n(ti -\)(n -2) 0''-^s\n - &c.
CIRrUI.AR FUNCTIONS. 337
Cor, Similarly may fe9" cos 6 be found and shewn to be
= 9" sin + n0''-'cose - n (n - I) 0""^ sin 9
-n{n - \){n - 2)9''-^cos9 + &c.
«^ T ^^ /. . , sin
75. Let -=«-.,„,<*=—;
rsin9 sin0 1 rcos9
■ * Jg IT " ~ (rT-'lYe"-^ "*" w -1 Je9"-'
/•cos COS 1 /-sin
ig "^^^ " ~ (w - 2 0)"-^' " W-2 "J, 0^
rsind sin0 cos 1 rsin0
sin cos sin ^
8ec.
{n-l)9"-' {n-l){n-2)9"-' {n-l){n-2)(n-3)9"-^
by which the integral may be reduced to / , (if n be an
^9 9
ri 9-9' \ ^ 6' ff'
integer) = h + kc.)=9 + 8ec.
"^ ^ Jq\ 2.3 2..S.4.5 ) 2.3' 2.3.4.5
a similar method applies to / -^^.
•Jq 9" •
76. Integrate sin m9 . cos w0, sin ?w0 . sin nO, and
cosm^ . cosn9.
Since sin ^ . cos fi = 1 . jsin (^ + 5) + sin {A - B)} ;
.-. sin m0 . cos w0 = ^ • {sin (m + w) + sin (m - w)0| ;
, icoslm + n)0 cos (m - n)9]
.♦, /,(sin m0 . cos n9) = - ^ . | ^^ '— + — ^ —\.
m + w m - n
Also since cosmO .cosnO = -| . jcos (m + w)0 + cos (w - w)0},
and sin wi0. sinw^ =\. {cos (rw - n)Q - cos (m + n)9\\
{s\x\{m + n)9 s\x\{m-ri)9'\
.-. L(cos m9 .cosn9) = ^ . { + >»
■'^^ '2[m + w m - n j
(sin(m + w)fc^ sin(m-w)0|
and /fl(sin m0 sin w0) = - i . •{ + }-
•"' ^ ^ [ m + n m - n \
CoR. Similarly if — - = sin (a + m0) . cos (6 + wO),
rf v
put for sin (a + m0) . cos {b + n9) its equivalent expression
i[sin \a-\-b + {m + n)9} + sin ja - /> + [m - n)9\\
22 2:
338 INTEGRAL CALCULUS.
77. Integrate (tan 0)"', and (tan^)""'.
(tan eT = (tan 0)"'-' \ 1 + tan^^ - 1 }
= (tan0r-''^-(tan0r-;
.-. /, (tan er = ^'Z^}]'' ~ ^' ^'^" ^^'""'
/, (tan er-^ = ^-^^3^ - fe (tan 0)-%
&c. &c.
.../.(.„.). =1^' -15^. <^«->-::'-.c.
by which w is reduced either to 0, or fg tan = - log cos 9.
'^^' ""^ iitanOr^ I (tan 0)'"
d . (tan 6)
'-'' (tan^r Jeitaner-' {m-l){tan6r-' J,(tan0)"
r 1 _ 1 r 1 ,
""""^ Je {tauOr-' ~~ (rn-S) (tan O)'"-^ ^^ (tan 0)-- '
. f ' ^ '
■' ^<,(tana)"' {m - l)(tan0)'"-'
1 1
+ &c.
(m - 3) (tan 0)'"-^ m-5 (tan O)*" " '
and thus u may be reduced to 6, or /^^ — = log (sin 6).
Jg tan (7
79. ^e'^sinA;ci7.
e'"'sin A-cT A; . ^^ , . ^
r^e^'sinfea; = . fX'coskx (l),
a a
e^'coskx k , „, . ,
and fe^'cos kx = + - . Le^'sin kx.
k
Multiplying by -, and transposing,
A;* , . ke'^" coskx A; . „, , ,.
%,.fe"'smkx= — — + -./,e""cosy^.r (2).
CIRCULAR FUNCTIONS. " ,'J39
Adding (l) and (2),
.-. f^e^" sin kx
k-\ r nr - . (a . sin kx - kcoskx)e'^
' ' " sin k,v = s— ■ — ' —
a
(a . sin kx - k coskx)e'"'
1^ + a
80. To integrate —
dx a + b . cos x
1-tan^-
bince cos .r = . Let ;jf = tan ;
^x 2 '
1 + tan" -
2
1 - z' . dx 4f«
.•. coScr = ^i .-. sin.r
14^^ dz (l + zy
dx 2
z^'' ' ' dz 1 + z^
du I 2 2
d^^ ^1 -2;'TT^"a(n-««)+6(l-xr^)*
Ti * • * / /1-2;S2 2;8
But sm cT = \/ 1 - ( 1 =
[l + zV 1 +
1 +z'
(1) Leta>6; .-, — =
dz a+b-{-(a-b)z^ a + b a-b .
^ ^ 1 + . z^
a + b
.-. w = -.V -.tan-'UV
a -k-b a-b \ a + b
2 , ( /a-b o!]
= tan-' -^3) &c., take the logarithms and differentiate;
mwsina? ^i sin a? + 2^2 sin 2ir + 3^3 sin 3a? + 8ec.
* * 1 + w cos a? Aq + Ai cos a? + J.^ cos 2 a? + &c.
Then •.- sin ,t cos aa/ = ^ {sin (a + 1) a^ - sin (a - l)a?}
cos a? sin j3a? = J {sin (j3 + 1) a- + sin (/3 - l)a7J ;
therefore multiplying out and arranging the terms according to
the sines of the multiple arc ;
= (^, + n - m Aon + — A.^n) sin a;'
2 2 2 2 '
. ^ 2 4^4 m ^ ^ . s .
+ (3-43 + - A.)n + n A^n + — A.7i) sin 3,r
2 2 2 " 2 ^
+ &c. ;
_2ww^o-2Ji (m-l)JiW-4^2
(m + 2) w ' ^ (m + 3)n
(m - 2)^2^ - 6-^3
{m + 4)w
hence if ^0, A^ are known, the other coefficients are also
known.
86. When w = - 1, or
A = 1 + -w + n^ + &c. = ^ „ ;
2 2.4 \/l - w2
113 2/ 1\2
^, =2n(-- --^w'-&c.):=-. 1 --^ - =-(1-^),
7/2 = - - . (w^„ + A^); A,= --. (Ai n + 2A.^ ;
y/, = - -(A,n + 2A,);
n
CIRCULAR FUNCTIONS. 343
similarly we may find the coefficients when w = - ^, m = - i];
the latter case is useful in Physical Astronomy.
du
87. Let — = log (1 + w cos I??).
loo(i -\-ncosx) = n co^x- ^rr cos^a? + ^n^ cos^ x-^ri^ cos^r + &c.
&c.
22 2.44 2.4.6()
3n^ 3.5 n' „ ^
+ (n + + — - — + &c.) cos w + &c.
4 3 4.G 5
= - A^ + ^1 cos.a?-.^2Cos2.r+ -(43Cos3ti' + &c. (l)
, , m^ 1 .3 n* 1 .3 .5 n*^
where J^ = + + . — + &c. ;
22 2.44 2.4.6-6
dA, 11.3 , 1.3.5.^ 1
dn 2 2.4' 2.4.6" ' y/i _ ^2
dn ny/i-n^ n n n
and C = log 2, for A^ - 0, when w = 0;
■-'='-r-^^^i
1 . 3 w^ 1 . 3 . 5 w^
and A. = n+ . — + — . — + &c. ;
2.4 3 4.6 5
dA
Ay 2 in^ 1 .3 n* 1.3 .5 n^ \
n n^\2 4 2 4.62 j
, n^y/l - n' n'''
w \ n I
and to find ^,5 ^3, &c. differentiate (l) ;
w sin X
Ay sin.r' -2^/2 sin 2 ^r +3 .,^3 sin 3*- - 4 y^^ cos4 ,r + &c.
1 +«cos i,(^^^^ = '"^nM^^^^'' 15(cos0r ^ 15.cos^r
^^^^ J, (cos BY (cos0f^^ ^*
.(sinf^)^ 1 |(sini9)' 4 (sing)"' «)
^^'^ 7,(^^70^-"^ 3^3 3|-
HcosOy f, ^,3 Scos^l 1 3 , ^^
^'"^^ X(si» ^)'- (^«sc>)' l2 (cos0f 2) sin
1 3] 1
3 , /TT
- . log tan I — +
,4 2
8
/,.>. r L_ = ^ ---cot 20.
^^^^ J, (sin ey (cos 0)^' 3 cos (sin 0)^ 3
( 1 fi) /, ( tan 0)' = 1 tan^* 9 - tan + 0.
i'i-r\ r___L__ - ^- ~ + + log (sin 0).
('^^ .',^t^ir0f "4(tan0y^2(tan0r °' ^
(18) /e0^cos0 = 0='sin0 + 30"-cos0 - 60 sin - 6cos0.
c w' (sin-'*')" ^v\/i-a!^ , x^
ti4>6
(21)
(22)
EXAMPLES.
f ~ tan -^a! = x tan " '<» - J (tan ~ \vy - log \/ 1 + x'
X.r'tan-' V - =— tan"^ V -
a 3 a
\/ax (or a x _^
3~ \5 3 '^^
(23)
/.cVcoo/..- ^,^^,
(24)
e""^.(a sin kx + k cos kx)
(25)
^ „,. ^ . _ e"^. sintt?(asincr-2cosaj) 2 . e'"
•'^" -v— y ^2^4 ■ „ (^a ^
*)
(26)
/* ^ ^ t.n-'f/vA. n^i
J,acos^0+6sin^0 vA&-'"' l^a'"'^)'
(27)
r cos9 1 sin
Jg (1 - e' cos^ 0)^ 1 _ e2 • ^1 _ g2 pos^
(28)
/• 1 1 f - 6 sin a? r 1
/ ,^ r, = -:^ — r,{-^ + « / ;
-1
(29) fe (cos 2 0)^ cos = 1 sin (3 + 2 cos 2 0) \/cos 2
+ - — 7= sin"' (v/s sin 0).
8^/2 ^"^ ^
(30)
(31)
sin 9
Jq v/sin^a - sin-0 ^ ' Vcos + x/sin'^a - sin=0/
from e = e\
j^^^Acosx^ (where A = log a), between x = 0, and .x = 7r
CHAPTER VI.
APPLICATION OF THE INTEGRAL CALCULUS TO DETERMINE
THE AREAS AND LENGTHS OF PLANE CURVES, AND THE
VOLUMES AND SURFACES OF SOLIDS OF REVOLUTION.
88. We have seen in the Differential Calculus, that if
y =^ f(,i) be the equation to a curve, and A the area of a
dA
portion ANF^ that — - = y = /(-»)• Hence, when the equa-
tion to a curve is given, its area may be found by finding the
value of fxfipo)^ and this integral may in general be found by
means of the rules given in the preceding Chapters. If the
equation to the curve be between polar co-ordinates, then
It is sometimes convenient to substitute z for (x) ; but
then, since y = f{%),
dA dA doj dai
dz dec dz dz '
. da; , ^^ ^ dx
89. Again, if s represents the length of a curve, of which
the equation is y =f(x),
dx dx~ Jr
dy'
where — may be found from y = f{x).
90. Also, if V and S respectively represent the volume
and surface of a solid of revolution, since
dV „ j^*^ . ./ ^2/*
—- = Try; and — = 27r?/ V J + ;7-i
dx dx ax
'. F = TT jxy"\ and »S' = 2 tt
/^^v^-S-
91. A constant must be added to each of these integrals,
the determination of which depends upon the nature of the
particular problem.
348
AREAS OF CURVES.
As an illustration, let the
area ABD be required, the na-
ture of the curve JNP being-
known by the equation 2/ =/(.r),
where AN = w, and NP = y.
Let AB = a, and ANP = A ;
dA
-T- =y
ax
/(^');
.-. A = ANP = fJ{.v)=c{>{w)+C (1).
To find C, we observe that if a," = the area = ; if there-
fore at the same time (p{x) = 0; .-. C = 0,
and ANP = (p (a?), and ABD = (pia) ;
the same result as would have been obtained had we succes-
sively put cr = and x = a in equation (1), and subtracted
the former result from the latter. This process is called
integrating between the limits of a? = and x = a, and is
commonly represented by the symbol f"f{a!) ; the first limit
being placed below, the second above the sign of integration.
To take a second instance, let the area DBCE be required
where AC = h; putting a for x in equation (l),
area ABD = (p (a) -i- C,
and area ACE = (p{b) + C;
.-. area BDEC = (p{h) - cp (a).
Hence, if the value of an integral u -^
.U9
A RE A.S OF C;U UV ES.
92. To find the areas of curves, or to integrati
dA _ . ^_ !l'
Ex. 1. To find the area of
the circle.
CN = .f
CA = a}
.-. A=fry = /, v/«' - .^' ;
.-, area C BPN= f,\/a^ - (v\
But v/«'-^'= /^ 2
= — sin ^ - + - V a'' - .r + C.
'2 « 2
C = 0, since area = ; if .r = ;
sin~^- can only be approximated to, by means of an infinite
a
series, but if .r = a, it = - , and,
quadrant ACB = T \/a' - x"- =
a' TT Tra~
2 ' 2 ~ 4
area of the circle = ira^.
Cor. 1. If AN = a', y = \/2 a 2? - ct?-,
and when tt? = «, JiVP becomes a quadrant ;
~ 4
•. f V^2rt.'
The two definite integrals j^\/d^-x\ and £\/2n.v - .v^,
should be carefully remembered.
Also f^\^ar - .v^ being = CBPN, where CN is some-
times called the cosine to radius CA ; .-. /,\/a"^ - x^ is called
350 AREAS OF CURVES.
a circular area of which cosine = x and radius = a ; and
Ly/^.ax - x^ in which AN = ^r, is called a circular area, of
which ver. sine = x and radius = a.
If AN = the diameter, the area ANP is a semicircle ;
-2a
... / v/.
ax — X =
Also '.■ ^ " \/ a? — x^ = area of the second quadrant;
••■/_/«'-- = -■•
Cor. 2.
To find the area of the sector AC P.
Let J = areay/CP; 9=^ACP',
•••^ = 4^-4"^
a^O ad radius x arc
.-. ^= — = ax — = — .
2 2 2
Ex. 2. To find the area of an ellipse.
The centre the origin ; CN = x ; NP = y.
.-. y = -w a- - x^
a
.'. A'^ f^y = -. f^\/a^ - x^ ;
J . ^ r" /-i — i ^ '^^^ '^"^
elliptic quadrant = - / '\/ a^ - x" = - . = ;
.-. area of ellipse = -rrab.
Had the vertex been the origin, and AN = x,
elliptic quadrant = - \/ 'iax - x^ = = .
^ ^ a Jo ^' * 4.
Ex. 3. To find the area of the common parabola.
y^=^mx\, .'. y = 9.\/mx.
A = l,y = 2 f^\/mx = 2 y/tn . f x^ + C ;
'. area = j y = x^ = ^2 \/m x .x = ^yx
= ^1 of circumscribing rectangle.
AREAS OF CURVES. 351
Ex. 4. To find the area of the Witch.
y = — y/ 2aoc - ai'\
ja = Ly = 2a / "^ = 2a / .
= 2a\ ( .^ " '^ + a /*-— ==l
I ^ T \/2aaj - oe' -^ x \/2ax - .v^)
= 2w\\/2ax - X-+ a ver-sin"'-^ + C.
area = 0, if .r = ; .. C=0;
•. area = 2a{\/2a,v — x" + a ver-sin~^-> .
I a]
Ex. 5. Find the area of the hyperbolic sector CAP.
Sector CAP^A CNP - area AN P. .
Let CN=x\ .-y
NP=y\; .-. y = -y.7^^. ^^^^ /
\ a ^ 1
CA = a] c A
b , y-r: b /- a?^ - a^
ANP = f.y = -/V^^- «^ = - . / y- ,
a a -^rVX^ - a
= -. I- v/^^T^^ - - . log {x + x/Z37)]+ C,
a (2 2 J
andO= loga + C; \-ANP=0; if.t? = a;
... ANP=-^ .log
2 2 '^ \ a /
6a , /a? + V.r^ - a\
= ACJVP--.log( ^ )
ba , Ix y\
sector C^P=-. log (^- + -j.
352
AREAS OF CURVES.
Ex. 6. Find the area of the
portion PNMQ, PQ being an
arc of the rectangular hyper-
bola.
Here yx = — . Let CN = a,
and CM = /3,
2 J.r.v 2
C;
... PQil/iV = ^" . (log ^ - log a) =*| . log (^) .
Also ••• ^ = -; .-. A CiVP= A CQM;
2 4
.-. sector CPQ = area PiVJ/Q = - log [-
Ex. 7. Find the area of the cissoid.
Here y^ =
2a - X
y =
^^y^LzTh
r'\/2<
= - 2 V 2 a - .r . .r^ + 3 . /^.ra \/2 1
= - 2a;\/2a.i7 - a;^ + 3 . j^\/'2.acc - .^•^
r-" 7ra^ 3 7r „
area = / « = 3 . = — a .
K 2 2
Ex. 8. Find the area of the cycloid.
dy \/2ax-oc^
Oriffin from the vertex, — - =
" dx X
dy
ea = l^y = yx - fx^— = V'V - jlv^nx - x\
also ••• if a; = 0, y = ; if .t? = 2 a, y = -n-a;
r'^" • , • 1 ^ 2 1 2 ^'ra-
•. / y = semicycloid = 27ra -^Tra = ;
•. cycloid = 3 TT a- = 3 . area of generating circle.
AREAS OF CURVES. 353
Ex. 9. Area of the conchoid.
Here xy = {a + x) \/b^ - a;^,
"'■^" = "" •"« ■ Lt;^') ^ (" ^ i) ^'"^
Ex. 10. In the common pa-
rabola, to find the area ASP.
SA = a, lASP=e, SP=r;
2a a
•. r =
1 + cosO .,0'
^=i
Jecos"- JflCos"-
= a^{tan - + - tan^ ->.
Ex. 11, Find the area of the lemniscata.
Here r^ = a- cos 2^;
• '. fe^r^ = -• fe cos 20= -sin 20+ C.
Area = 0, if = ; .-. C = 0; and area = — sin 20.
4
Let = 45"; .•. 4^th of lemniscata = — ; lemniscata = o^.
Ex. 12. Find the area of the spiral where r = a0".
dA /^•^»
Here -—r = i ri
^'^ d0
-2' '
a —
\a} '
d0
dr~
1 i
na"
-'. ...
dA
lb'
1
2 no"
i+-
.-. A =
2na'
2 W + 1
2n+l
+ c,
and C = 0, if J = 0, when r = 0.
23 A A
354 AREAS OF CURVES.
CoH. Let n-=\, or the spiral be that of Arcliimedes ;
.•. area = — = — ^- ,
6a 3R'
if i? = r when = Stt;
therefore area of spiral in first revolution = .
The area after two revolutions of the radius vector is
when 6 = 47r, or when r = 2R. But before r = 2E, it will
have made two revolutions, and therefore have twice gene-
rated the area from r = to r = R.
Consequently we must subtract the area described in the
first revolution from that in the second ;
7r.(2i?)^ ttR' _ lirR^
■"■ ^'^^" ^ 3 ~ 3 '
And the space between the arcs of the first and second areas
3 3
At the n^^ revolution r=7iR,
{n - \y^ r = {71 - 1)R;
, . TT (nRf -(n- ifR'
.-. area after n revolutions =- .
3 R
irR^
Area after (w + 1) revolutions = —^ {{n + if - w^} ;
3
.'. space between the arcs after ji + 1 and 9i revolutions
= n times the space between the first and second.
Ex. 13. Find the area of the involute of the circle,
where r^ - P^ = ^'
clA _ J ^de ])
dO ^ dr r\/r"-p"'
dr ~ 2\/r'- - p'^ 2a
AREAS OF CURVES
S55
6 a
+ C ; and C = 0.
If p = ^TTff ; A = ^ir^d\ or subtracting iro^, the area of
the involute exterior of the circle after one unwrapping of the
string
Ex. 14. Find the area of the curve of which the equa-
tion is
y^ - 3axy + w^ = 0.
If the curve be traced there will be found a nodus as
JPMQ, to which the axes Ay and Ax are tangents.
y
Let y = xs;;
.'. at^z^ — 3n,v^% + x^ = ;
- = ^ = tan PAN ;
V
3az 3a
X = r, , and y -
1 + z"
and since x is = 0, for each of the branches APM and AQM,
this will happen if ^ = oo or = 0.
dA dA d,v dx
Now — - = — — ,—-=«.-—,
dz dx dz " dz
dx 3aA\ +z^ - 3z^\
and — = ^^- — ^ = 3a
dz (1 + z^y
9.%^
^ , dx , rz'{\-2^)
.,A = j:y.- = 9ar
(1 + zy
rz~ [^i — "z^)
V, (1 + ^0'
•'Ma {i^-zy \
AA 2
356 AREAS OF CURVEvS.
Let ;:^ = 0; .-. C = ^, and let %, = ^- at 3/;
2 .r-
So- r ^ 1 2 1 1
.-. area AQMm = + 9a"' { - * • 7 377, + q> •
2 ( ^ (1 + z.y 3 1 + z;'^
Integrating j:r = co and ^ = ^1 for the branch JPM,
area APMm = 9a- < - | — + - . -oj ;
i ^ (1 + z^y 3 1 + z^^
3a^
.-. the nodus APMQ = area APMm - area AQMm = — .
If the area of the nodus only be required, the following
method is often useful ;
Since tan = - = z ; .•.-—= cos^ 9 ;
w dz
dA ^ ^dd , „ „,, *'•-
dz '^ dz '^ 2
Here
AREAS OF CURVES. 357
.-. ^ = 3aj3.^sin'0cos''6>; the limits of which arc 6=- and 0=0,
since those of x are, and a.
t'os sin' 6 1 r • ,• /I
Now fg sin' e cos- 6 = -^ - fg sin" ;
cos 6 . sin^ 6 , , , /^ 1 A '^
but = 0, both when 0=0 and 6 = - ;
5 2
1 5.3.1 TT
.-. /J' sin 6 cos- = - 1 /^' sin" 9 = - - x
.'. A = 3a(i.
6.4.2 2
3uj3 TT
() X 4 X 4 8 4
, , , 3 7ra/3 3 (a^ - 6^^
.-. ^A = whole area = = - tt
8 8 o6
CoR. For the evolute of the hyperbola, where f-J - (7;) =^:
make - = sec' ; .'. ^ = tan' ; ■• -^='^a sec' . tan 9 ;
a p "0
, rsin* 9
.-. A=3a^.fo sec' . tan'0 = 3a(i. — ,- ;
'^ ■' J cos
the limits of 9 being 0, and some finite value 9'.
Ex. 16. The same substitution applies to find the area of
For making a? = a cos2"+' 9; .-. y = ^ sin'^"^ ' 9,
— = - (2n + l)acos^"0 sinO;
d9
.-. A= - {2n + l)a/3 /; sin^"+=^0 cos^^" 9
= - (2w + l)a/3/^{sin -"+20(1 - sin-6>)"J ;
2
(2w-l).(2w-3)...3.1 TT
2«.(27i-2)...4.2 '2'
1.3.5. ..(2w + l) 1.3...(2w + 3)
whence expanding, •.• jl?,m^"9
^ _ [1.3.5. ..(2W + 1) 1.3. ..(2w + ^
.-. area = 4 J = (4w + 2)7rap.{ ; - n .
^ ^ ^ (2.4.6. ..(2W + 2) 2.4... (2w+^
(n-1) 1.3...(2w + 5) (w-l)(w-2) I.3...(2w + 7) ,
Jt-n.- . n . ; r + &t
2 2.4...(2w + 6) 2.3 2.4...(27i + 8)
358 AREA8 OF CURVES.
If w =1, l^] + (-) = 1, and area = fi-TrajS | — :.>
= - 7ra/3, the result obtained in the preceding example.
8
THE LENGTHS OF CURVES.
93. To find the lengths of curves, or to integrate
ds / dy^
dTv^^'^d^^' when 2, =/(..).
Ex. 17. Find the length of an arc of the parabola.
dy 2m dy^ 4!nr m
ax y dx^ y x
J^ dx' J:c X J^ ^/a;
m m
^ x+ — , —
r X + m j 2 j 2
"^'■^ \/ x'^ + mx Jxy/ x^ +mx Jv \/x' + vi x
= s/x^ + mx + — low (x+—+ \/x- + m x) + C.
2 2
And 5 = 0, if cr = ; .-. = - log - + C ;
2 "= \2)
,-z m , {2x + m + 2 \/ x- + mx\
.'. s = %/ x^ + mx + — log — .
2 ° V m I
Ex. 18. Find when curves included under the general
equation y = ax" are rectifiable.
dy m "^ r ^ / m-d~ '^^^^^^
-^ = —ax " ; .'. s = I \/ \ + --„- . A' " ;
dx n J.T rf
which is integrable,
n
(1) When is an integer = r,
^ ' 2m -2n ^
m 1 ml 2r + 1
or \ = — , or — = — + 1 = .
n 2r n 2r 2r
LEN(iTlIS OF CU11VE«. 359
(2) When + i = an inteoer = 7,
m 1 m 2q
or 1= . or — = -~ .
n 2q - \ n 2q — \
Let T = I, 2, 3, &c. 7=1, 2, 3, &c. ;
m 3 5 1 in 2 4> 6
•••- = -, - , - , &c. and — = - , - , - , &c.
w 2 4 () n 1 3 5
m 'S
tj\. 19. Let — = - or the curve be the semi-cubical
n 2
parabola ;
, dy 3a . V^' ,- 2
dco 2 v/c 3a
If 5 = 0, .2? = ; .-. C = - — 7:^ - C5 ;
v/c3
Ex. 20. Find the length of the cycloid.
dy . /2a -ce dif 2a - x 2a
^ = V ; ... 1+^ = 1+ = _;
ax lV dcV' 30 X
^x X J^ ^ X
and C = 0, since s = when x = Q>\
therefore s = 2 v 2 ax = twice the chord of the arc of the
generating circle, corresponding to the arc of the cycloid.
Hence the cycloid is rectifiable.
If X = 2a, * = 2\/4a^ = 4a, or the length of the semi-
cycloid = twice the diameter of the circle.
Ex. 21. Find the length of the arc of an ellipse.
6 /- dy b X
y = ~\/ d--x-\ -f- = . —
^ ' dx a v a' — x~ '
dif ^ b^x- a* - (a" - h-) x^ a- - e'-x"
dar «^ (a" — x-) d~ (a~ - iV^) a ' - x'
360 LENGTHS OF CURVES.
.'. s= - , = a / , (if .V = za)
-^s ^a' - ar" -^z v/l - z'
J \/l-2;^ ^ 2 2 4 2.4.6 '
the integration of which depends on J / ^ •
If the quadrant be required, we must integrate from
a? = to /l - e' co9^0
= ajl -le^cos'^a- ^-^e^os^e- ^-^^e^cos'^0-&c.r
' ^ 2.4 2.4.6 '
Now /ecos^"0 = sin . cos-"-'0 + {2ti - l) - /^ cos'-"-'0 . sin^O
sin 0cos'^"-'0 2w - 1 . ^
= + . Lcos-"~~6,
2n 2w •'^
and sin 9 cos"" '^9 = 0, when 9 = (>, and 9 =- ;
2
.-. calling A cos-" = Po„,
LENGTHS OF CURVES. 361
p. =^-^ p. _(2n -l){27l-3)
~" 2n * '""' ■ 2w (2w - 2) '""'
_ (2?^ - 1) . (2n -3) 3.1 TT . p _^
2n.{2n-2) 4.2 '2' " °~2
T« . 1 2 1.3 , 1.3^5 ^ 1.3-.5-.7
2 ' 2- 2^4=^ 2^. 4-.fi'- 9=^.4.2 fi^oa*^ «^-J-
Ex. 22. Find the length of a hyperbolic arc.
b , dii h X
y = - s/oo' -a\ ~ =~ ■
a ' d,v a ^ g? _ fj,}
" dx ^ ^'^ do!''^ a^ (^v' - a^) ~ ^ of" - d' '"
= ae % ______
X- v/;^=^ - 1
_ r ez i J 1 1.1 1 1.1.3 1
^'"'I'x/z^i ~^(^~ 2T4'(^^" 2.4.6* (^«
1.1.3.5 1 1
. -&c.>,
2.4.6.8 {ezf )
the limits being .r = a ; .r = co , or ij? = 1 ; xf = cc ;
now every term, except the first, depends upon
f . , 7n odd,
, r 1 1 v z'' - I m- 2 r 1
and / , = — . -— — + . / 7=^^
-} z'" \/z^ - 1 irt- 1 x'" ^ m- \ J.^z"'-~y/z' - 1
and . vanishes both when ^ = 1, and ^ = x ;
362 LENGTHS OF CURVES.
r'^ 1 m - ^ r'^ 1
Init / . = sec~^ z = - ;
^i z y/z' - 1 2
-1 ;^V^=^-1 ^ ^1 ^'\/^'-l 2.4' 2
2
1 .3 TT
1.3.5 IT
2.4.6 2
1 1.11
Z' iC TTU, 1 1.1 J
1 . 1 . 3^ 1 1 . 1 . 3^ . 5^ 1
"^ 2^4^6 ■ ? ^ ¥7^\&~8 • 7 "^ '^''''^-
Now the equation to the asymptote is y = — ;
.'. leno;th of asymptote = 'V *' + - o ='^' V r~ = <^*' = «'''^-
'^ "^ a" a''
But ae / — , ^ =ae\/z^ - 1 = ae;^ from ;? = 1 to ;^ = w .
If therefore I be the length of the asymptote,
a , J 1 1.11 1 . 1 . 3^ 1
2 t2
I - s = ]-k-- + — - • -3 + ^ — t-t:
1 . 1 . 3^ 5^ 1 ,
2■-.4^ ()'.8 C *
Ex. 23. Find the length of an arc of the logarithmic
curve.
dy
Here y — a% and — = Aa^ - A . y,
dx
ds ds da; > ., 1
and — = — . - = \/\ + A'y . ;
ay d.v dy Ay
LENGTHS OF CURVES.
363
r ^/i + AY ^ r _Jy^ . ^ r L._
> Ay J,v/i+^V -^'Ayx/l+AY
v/l + AY 1 ,
Ay
+ C.
A ■ J ^ 1+ v^l + A'f
Ex. 24. Find the leno;th of an arc of the Lcmniscata.
r^ = a^ cos 20, anc
= / . = « . / --.—.^^ , if r = az
1.3, 1.3.5
^'"■+&c.}.
4 2.4.6
Integrate from = 45" to = ; or from z = to ^ = 1
' ^2
/-l 1 TT r' Z TT
/ . .- , and / — -=
'o v/l - ;J?- 2.42 Jo \/l
1.3.5
V
Tra
4.6 2
2 .2 ,2 q2 .2 ^2
1 l" . 3' r.3\5' r.3\5\7
r.+
,-&c.},
2' .4- 2^ 4'^ . 6'^ 2\ 4^ 6^
The whole length of the lemniscata = 4s.
Ex. 25. To find the length of the involute of a circle.
ds r r
Here r^ - p^ = a- ;
dr \/r''' — p^ ^
2a 2a 2a 2
If = 27r, or the string be unwound once, * = 2 7r^a.
If = 2M7r, or the string be unwound n times length
= -(2W7r)^
2
Ex. 26. If the radius of the circle be unity, and be a
364
LENGTHS OF CURVES.
circular arc, 0, the length of its involute, O. the length of the
involute of 0], &c., prove that
e + e, + e, + Sic. = e'-i.
a
JP, = 0,.
Pp ^ l\lh
QP PP,
Then APQp = LP,Ppr-,
.. de, = -^de, = \B'de-.
V
0'
And de., = fde.. = e...^ = ^^.d\de
.-. e.
6'
2.3.4
.-. e + o, + 0, + &c. = + 1 0- + -^ eu &c. = e' - 1.
THE VOLUMES AND SURFACES OF SOLIDS OF REVOLUTION.
94. To find the volumes and surfaces of solids, or to
integrate the functions
dV ., , dS ^ / df
= Try\ and = Svry V 1 + . ., •
d.v
rf.r
d.v'
VOLUMES OF SOLIDS. 3G5
Ex. 27. To find the volume of a cone with a ciicuhir
base.
Let a = altitude, b = radius of base.
Then if the vertex be the origin and the altitude the
axis ot .p, y = - .r ;
a
And F=0 if .r = 0; .-. C = 0; .-. F = - . - .
a^ 3
Let tt? = a ; ••. whole cone = — — = 4 of a cylinder of
3 -^
the same altitude and on the same base.
Ex. 28. Find the volume of the paraboloid.
y^ = 4»»' - .'P' - f ijr - .^0^},
and.r = 6; 7^=0; .-. = C - tt Lft^ ^| ; .-. C = '^";
. • . F = — ^ TT I « />- sni - ' - - ~— ^— - (x' + 2 /y-') } .
Let .T = 0: .-. wl
1 1 r'i"^" -''\
lole volume = 7r/r < — + — > .
I 2 3 /
VOLUMKS OF SOLIDS. 36?
Ex. 32. Find the volunic
generated by the revolution of the
cissoid round its asymptote.
AB= 2 a, BM = w, MQ = y.
or a?' =
y
^±zJ!}\ j.....^^
But x^y- = yi^a-yf; .-. xy = \/y .{2a- yf- ;
••• /y'^y = lA^^-y) V^^ny - if
!y (« - y) v^2 ««/ - r + « !,j \^^ay - «/'
(2ay-y")^ r / x
^ ,^ - + af^x/2a.y-y';
•. F = TT {(2«7/ - f)^ - f (2af/ - f'Y^ -2a fy \/2ay - y'\.
whole solid = / y~ = tt .2a . — - = tt"^ a'^
2
Ex. 33. Find the solid generated by the revolution of
the semi-cycloid round its base.
dy yj 2ay - f
Make the base the axis of x\ .. -— =
y
dx
dy " ' ^.,\/2ay-y'
V = Liry-.-r =71". / — >
r y y'- \/ 2a\i - y 5 / — —
J\/2at/-2/' 3 2.3 "^
3.5 „ / , 3.5 .
a' V2aw - y~ + a ver-sin
2.3 2.3 a
... r=,/
'\/2ay-'f 2
Ex. 34. Find the solid generated by the revolution of
the cycloid round its axis.
and y = « (0 + sin Q) ; x = a {\ - cos 6) ;
368
VOLUMES OF SOLIDS.
.-. V=7rn' 1,(9 + smOy sine
= '7ra^fe\e''sine + 29 sin^ 9 + sin' 9] ■
whence intea-ratino; from 9 = to 9 = tt ;
-r-^-g
Ex. 35. To find the volume of a conical figure, the base
of which is bounded by a given curve.
From A draw JD perpendicular to
the base, and = a. In AD take AN = x,
N being a point in a section 6c, parallel
and similar to the base BC
Let A = area of the base,
S = area of section be ;
S _hN-' _ AN- _ a;\
■' A " BD"" ~ AD' ~ ^'
S = A —•> and ^—
a dv
S = A
x'
A , ., Ax' ^^ , ^
— r .v' = + C, and C = ;
ABC =
Ad'
3d'
A . a
3
base X ^ of the altitude.
CoK. This proposition is manifestly true for a pyramid
of any base.
Ex. 36. To find the volume of a Groin ; a solid of which
in this instance, the sections parallel to the base are squares,
and those perpendicular, bounded by a given curve.
Let the given curve AD be a quadrant
AN=x, NP = y; AB ^ BD = a .,
therefore generating area = (2?/)" = 4y/' ;
47- = ^{2a.v - .v');
d.v
= 4«% if .V = a.
Again, •.• generating surface = perimeter of square = 8//;
dS
VOLUMES OF SOLIDS.
ds
tm
dx
S = Saw
8y — = 8 \/a'
dx
80;
And similarly may the volume and surface be found,
whatever be the curve APD. Also, if the base be any other
figure, of which the area is a function of y, as a circle, a
parabola, a triangle, &c and JPB be a curve of which the
equation is y =f{x), the surface and volume may be found.
Ex. 37. Find the solid generated by a ^^-"f
parabolic area round its ordinate.
AM = a?, BN= X,, AB = a,
MP^^, NP=y,, BC = b;
dv „ . .., ( y'
dx.
= Try\
(a - xy = TT a
4>ma — y'
(4/7^)
4rw
:(b'-y')
dV _ dV
dxi dy
(4 m)-
{b'-2bY + y'
(4m)M ^ s 5/'
vol I
■b'
1 -
i)
Trb" 8 8 „,
— = — Tra'b.
(4r»)- 15 15
(4w)-
Ex. 38. Find the volume and
surface of the solid generated by s
the circle BQP round an axis ^
ANx, in its own plane.
Let AO = b, OB = a,
MQ = y, OM = X.
Then surface generated by QP
= TT {NP" - NQ') = 7r{{b + yf-{b- yf\ =4>7rby;
dV
= 4>7rby; .-. V = 4-7rb f^y = 4>irb -^ = 2'7r'
a b.
S =27r. [ANP + iVQ)-T^= 47r6. f—= ^irb .ira = A^ir^ba.
■'' dx 'Jx dx
24
Bb
370 SURFACES OF SOLIDS
Ex. 39. The surface of a sphere.
dy a - X
doc \/9,a,v - a?-'
y =^ \/2ax — aP'y and
dy' (a - xY
1 + ^ = 1 +
dai^ 2aa? - x 2ax-x- y^
S= STr/^yVl + -—-^ = 2 TT Jly .- = 27r f^a = 27rax + C,
S = 0, if X = ; .'. C = ; .*. surface of a segment = Zttux ;
.-. surface of sphere = 27ra .2a = 47ra^
Ex. 40, Convex surface of a paraboloid.
dy 2m
y = 4ma;, — - = — ;
dx y
dy^ 4m^ m x + m
.-. 1 + :r^„ = 1 + -^ = 1+ -
dx- y^ X
.r
? • /r \/^T
^. + m
= 4 7r Vm/^ v/.T + 7W = 4 7r \/m| (cT + m)^ + C,
and if ^ = 0, ^ = ; .-. C = - 1 tt \/m . m^ ;
.-. surface = ^^— . {(x + m)^ - m^| .
Ex. 41. Surface generated by a semi-cycloid round its
da? y f/.t;'" y
dS dS dx / y
.-. -— = -—.-—= 2 ttV 2 ay . ^ ^' -'^
dy dx dy \/2ay-y^
.'. S = 2ir\/'2a. f —-
''VV 2a-y
= 2'7r\/2a{- 2y\/2a - y {2a - y)Sj;
4 32
.'. surface by semi-cycloid = 2 tt . - (2a)~ = — 7ra''.
SURFACES OF SOLIDS. 371
Ex, 42. Find the same when round the axis.
mi , . . dy /2a - oc
The vertex the origin, ^— = \/ — — .
dx
X
Surface = "^ir iy —- = ^irlys- Ls—-\\ s = 2\'^2ax
dx (• dx]
= 2 TT jsi/ \/2ax - 2 \/2af,\/x \/ ~^ ~ '^ i
= 47r|.Vv2a.r- \/2af^\/2a - x\
= 4 7r\/2a{2'V^-^ + -\(~" - •'^)-('
from .X' = to A' = 2 a, or 7/ = to ?/ = 7r«,
6'= 47r \/2ff J7rfl\/2a - -. (2ffi)^(
4 > of 41
Ex. 43. To find the surface of the prolate spheroid.
b / „ , dy- rr-e-x'-
y =^ - \/ a~ - x^, and 1 +
dx"
dS ds b /- a--e'x~
— = 27rV — = 27r-\/a^-.T- V — ^ r
dx dx a a^ - xr
2,6 ___, . ^^
\/ d^ - e^x^ - 2 7rb
a a-
-.6^= . sin"'<^ 2>
= 27ra-|\/l - e- . + 1 - e^\.
sin " ' e
Let e = 0, or spheroid become a sphere; .•. = 1,
and surface = 2 7r«"^l + l| = 47rO'.
372 SURFACES OF SOLIDS.
Ex. 44. To find the surface of an oblate spheroid.
BM=x, CN=.v,
MP = y, NP = y, ^-^^^
dS ^ ds XT
dwj dx^ ' / 1
dS ds / i
or — - = STTiT . -— ; aZ_ J
dy dy X
dS ds /or - e^x'
.-. -- = 27rx - = 2 7r^f/" is
= ah+{aJ' -If) tan-' (^^
(6) The area of a parabolic segment, cut off by any
2
chord = - of circumscribing parallelogram.
(7) If r = a sec - , the area included by the curve, the
asymptotes and tangent at vertex = 4a^
(8) Find the area of x'^y'^ - a^y* = a*.
a^ { , /xy + a^ , I o?\^
Area = — -( log V -^ tan"' [ — ]}.
2 [ xy - a^ \xyj J
(y) In a parabola, find the area included between the
curve, its evolute, and its radius of curvature.
Area = 4 \/ '-{a^ + -ax + - x' } .
a\ 3 5 j
(10) If the subtangent of the logarithmic curve = that of the
spiral, 9 = - : the arc included by two radii of the spiral = arc
included by two respectively equal ordinates of the curve.
(11) Find the length of the spiral of Archimedes.
r ,~ a fr + ^r'
= — V r^ + a" + - . log
log
(12) The length of the epiycyloid after one revolution
of the generating circle = 8 - (a + h), and the area between
a
the epicycloid and the circle = irb^ I 3 + —
(13) Find the volume generated by the revolution of the
Witch, round its asymptote. Volume = 47r^a\
(14) The area of the curve in which
{a' - h^) sin cos ^ tt
r = —j-^^^-=^- ~^-', - \ is = - (« - h\~.
ya2sin^0+ forces-- 0' 2 ^ ^
CHAPTER VII.
DIFFERENTIAL EQUATIONS.
95. In the integrations which have been performed in
the preceding Chapters, the differential coefficient has either
been a given function of one of the variables, or else has
been expressed in such terms of the two, that by a very
evident process it has been reduced to a function of one only.
We now proceed to integrate differentials, when the differential
coefficients and the variables x and y are mingled together.
96. Differential equations are divided into classes, de-
pendent upon the o7'der and degree of the differential coef-
ficient.
Thus an equation involving
dy d'y d^y d"y
dw ' dx^ ' da^ ' dx"
m- (i:)'-©"
is called a differential equation of the w'^ order and of the
first degree, while one containing
dy (dy\" (dy\
dw
is said to be of the first order, and of the w"" degree: and finally,
an equation in which are to be found the n^^ powers of the
differential coefficients and the m^^ differential coefficient, is
named an equation of the m}'^ order and the 'nP^ degree.
We shall begin with that class in which the first power of
the first differential coefficient is alone found.
Differential Equations of the first Order and the first Degree.
97- These are included under the formula
dx
where M and N may be any functions of x and y ; we shall
however in the first place treat of homogeneous equations.
dy
98. Let J/ + iV — = 0, be a homogeneous equation, or
one in which t!ic sum of the indices of // and x together, is the
same in every term.
DIFFERENTIvVL EQl'ATIONS. 375
dy d%
Make y = xz ; .-. -— = z + ,v ~~ .
dx dx
Divide by N and the equation becomes,
M dy M dz
— + -— = 0, OV — + Z ^ X ~ = 0.
N dx N dx
M . y
But — is of no dimensions or is a function of - or
N X
Let .-. ^ =/(.); .....^-^^=-|.^/(.)S;
dx
log
xdz sr+/(^)' " UJ l^+f(^y
which may be integrated by the ordinary rules.
We put X =. yz, or y = xz, as may be most convenient,
for the solution is more easily effected, when we substitute
for that differential coefficient which involves the fewest terms.
dy
Ex. 1. Let X + y = (x - y) — .
dx
dy dz
Here make y = xz ; .'. — =z + x — ;
dx dx
dz X + y 1 +
.-. Z + X
z
dz I + z^
z
1
" dx I - z
z
dx 'V -y 1
dx _ 1 - «
xdz 1 + ij?^ 1 + ^^ 1 + ^"^ '
.-. log I - I = tan~'^ - log V 1 + ^-;
(X / tA , y/ x'^ + y^ «
•■• log(- Vl + JJ^' , or log-^ _^=tan-i-.
Ex. 2. Find the curve in which the subtangent is equal
to the sum of the abscissa and ordinate.
dx
Here y — = x + y ; and let x = yz •,
dy
dx dz X + y
dy dy y
ydz \cl y
\og{^ = z =
376 DIFFERENTIAL EQUATIONS.
Ex. 3. Find the curve in which the subnormal = y - .v.
dy
y
— '■p ;
dx
X
y '
Let
y = wz ;
.-.
•. z +
dw
xdz
dz 1
X— = \-
dx z
— z
~ z' -z + l"
z
z
,. log h/f-y-^-'\ ^ _L eot-(^n .
^ c / VS \xV3)
Ex. 4. Find the curve in which the distance from the
origin to a point in the curve equals the subtangent.
Here AP = NT, or y/y- + x'- = v — .
dy
liiT I ^^ v^-^^'^ + y" / T.
Make x = yz\, .'. z + y . — = = \/ 1 + ^- ;
dy y
dy
x/i
+ z- + z;
ydz y/l + 2^2 _
whence log {— ^ \ = - {x + \/ x' + f).
Ex. .5. {y/x - \/y) = \/y . — . Make x = yz.
dy
dy /
Ex. 6. x~ y = V A^^ + y^ • Make y = xz.
dx
Then a^ = c' + 2cy.
Ex. 7. X— -y = y log ( - ) ; .-. / = ^'e'.
dx ■ \xj
Ex: 8. Xt/ = ^'; .-. (x^-2yr = cx\
X
Ex. 9. NT and iVP are the subtangent and ordinate of
a curve of which the vertex is J, and tan TPA = m tan APN,
find the equation to the curve.
{rf + arY + ' = c-"' x\
Ex. 10. fA'^-y) = 'lf.ry^ ■■■ y'-'.^-'^r^"-'.
diffi:rkxtial kqi/ations. 377
Ex.11, fr(.vy') = I ^v f,f ; ••■ y' = CO,:
99. The equation (a + h.T + ey)dx + {ai + hiX + c,y)dy =
can be rendered homogenous by making
V = a + b,v + cy, and z = n^ + ft, .v + c^y ;
.-. dv =bd,v + cdy, dz = h^dx + c^dy ;
.'. c^dv - cdz = {bc^ - bic)d,
i.e. (a + 6 • _ 1 ^ y
101. 1 he ecaiation if" — + Py'" =z Qy" may be reduced
(Ix
to the preceding form, in the following manner.
DIKFEHKNTIAL lOQUATIOXS. 379
Divide by ?/";
d u
■■■ y"""~'-^l+ Py'"-" = Cl
,dy dz
^ ^ -^ dx dx
d z
.-. — + (m -n)Pz = Q-.
dx
which is of the required form.
^ dv hv~ m
Ex. 1. V = :^.
ds s s"
^ „ dv dz
Let v^ = 2z ; .'. v — = — ;
ds ds
dz 2hz m
ds s s~
Here P= - - ; .-. j,P = - 2h log (s) = log— ^^ .-. e''^ = —^ ;
= - m is-^~''^-'' = c + ;
^ 2h+ 1
= — = cs
(2h+])i
Ex.2. ^ + y = xy'; .: 1 = .t? + 1 + Ce^
dx ^ • y
Ex. 3. ^ + ^^^ =.r\/^; .-. x/y^C^yi-x'-Ul -x"").
dx 1 - x'
±iX. 4. xydy + y^dx = ;
3a C
•'■ !/' = ::—+ — •
2 .r- x"
Integration of exact Differentials. The method of Jinding
a factor which will render a function integrahle.
102. The equation Mdx + Ndy = is not always the
result of the differentiation oi f {xy) = c\ for after the dif-
ferentiation, its terms may have been divided by a common
factor, or the equation may arise from the elimination of an
arbitrary constant between the primitive equation and its
derivative.
380 DIFFERENTIAL EQUATIONS.
But whenever Mdx + Ndy = is the complete differential
. , , , ,. . d^u d^u
of a function of two variables, the condition — — = - — - is
dxdy dydx
du du
fulfilled, or since M = —- and N = -;
dx dy
dM _ dru^ _ dN
dy dxdy dx '
Hence, we have a method by which we may find whether
any equation of the form Mdx + Ndy = 0, be a complete
du du
differential ; and if it be, since then -— = 31, and 7- = iV,
dx dy
we can by integrating these partial differential equations find
u ; for since 31= — , 31 k the partial differential coefficient
dx
of u, with regard to x, considering x alone to vary, and its
integral will give all the terms in which x is to be found : let
the integration be performed. Then
u= j,3f+ Y.
Here instead of adding a constant C, we put F, for as y
has been supposed not to vary, the constant will include those
terms of the original equation which are functions of y alone.
Next to determine Y: differentiate with regard to y;
du _ df,3I d Y
' dy dy dy
du ^^ dY ^_ df^M
dy dy dy
103. Since Y ought to be a function of y only,
f I N ^ I should be independent of x.
Jy V dy J
To prove this, let y + ^y be put for y in /^i/;
... /, (i)/ + ~^y+ &c.) = J,3f + ^y f^^f + &c.
DIFFERENTIAL EQU.VTIONS. 381
dY rdM d'Y dN dM
= JV - / -~ ; .-. -,~^ = ^ = ;
J, dy dan
dy Jx dy dxdy dw dy
dY . . .
,— differentiated with regard to x vanish
dy ^
function of y only : the same result would have been obtained
1 the first instance.
2d.r 2a!dy
dY . . .
or since ,— differentiated with regard to x vanishes : F is a
dy
I only '•
by integrating N or — in the first instance
dy
Ex. 1. Let du
V it?* — y~ y s/ ob^ — y^
2x
Here M = -y ; N = -
V a?^ — y- y \/x^ - y-
dM 2y dN -2f - y' \ ^y
dy (a?- - y-)^ ' dcv y \{a"' - y~)y {x" - y'^)^
.-. u = j^M + F = 2 log {x + \/a?- - y^) + F,
du - 2y dY -2x
dy {x + \/ or - y') \/ x- - if dy y s/cF^^~
dY 2 f 2/ *'l
" dy~ ^cc' - f \x + \/x^ - f m
- 2 ^x- - y^ + X \/x^ - y^\ - 2
y'
. F = C - 2 log y
- 2 Ix- - y^ + X \/ x" - y ] - 2
\/x^ - f I y (^ + s/sc' - \f) \ y
X ^ s/x" - y-\ -
•• M = log ^ + C.
„ ^ , a(xdx+ydy) ydx-xdy , , ,
Ex. 2. Let du = ^V^ ^1-11 + ^ JL + 3hy-dy=0.
y/x' + f ^v' + y'
M = -y-^=- + — ^^ — ; N = , ^ - TT + 3by-.
\/x' + f- 'V' + y' . \/x' + y^ .T- + y-
dM -ay x" - y- dN
dy {x' + y-)i {.v' + y-y ~ dx '
.-. u = LM + Y = a s/x^ + y^ H- tan"' - + Y,
y
S82 DIFFERENTIAL EQUATIONS.
ay X dY dY
dy ^/.v^ + y' ?/ + ^- dy dy
.-. Y = hy"^ + C, and u = a v-t?" + y' + tan"' - + by^ + C.
.„ rxdy - ydx . x
Ex. 3. / - ,— V- = tan-i - + C.
'^ x^ + y y
^ dcz- w^da? «d« dw wda? - xdy ,—_
Ex. 4. _ + ^---^+^+^^ ^_^y/.^-~'+^-'=0;
.-. log (,7?2/) - - (?/ + n/.-p- + y"-^) + log (if/ \/.t- + y^ - y-) = C.
104, When the equation Mdx + iVd^ = does not fulfil
dM dN . .
the criterion of integr ability. —~— = — — , it is no longer a
dy dx
complete differential, some factor having disappeared from it.
Could however the factor be restored, every equation of this
class might be integrated by the same process : but there is
great difficulty in finding this factor ; in most cases the dif-
ferential equation, by which it is to be determined, is more
complicated than the original one.
Thus, suppose x to be the factor, then Mzdx + Nzdy =
is a complete differential, and therefore
d{M^) _ d(Nz)
dy dx
dM ^d.% dN dz
dy dy dx dx
whence % is to be found, a problem seldom practicable.
105. When, however, any factor of the equation
Mdx + Ndy =
is known, an infinite number of factors may be found which
will render the equation integrable ; for let z be a factor,
.-. du = zMdx + Nzdy ;
.-. {u) du= z-
■9)}-
= - e
;^ . sin g- .
cos;
^""^Ttdt
d0^
df
e' (sin'0 ■
- sin
'0)
= - . (cos
2
20 - cos
29)--
~ 2
. f cos (p +
9)-
cos (jt> -
-<,)]
= - e^ sin
p sin 7 ;
d^p dp dq cos 7 d^q dp dq cosjo
df ' dt' dt sin 7' d^^ " dt dt sinp
DIFFERENTIAL EQUATIONS. 393
d 1^] d.l^)
\dt 1 cos q dq \d^J
cos p dp
dp sin q ' dt^ dq
di dt
sin p' dt^
' ' ^°S I j" I ~ ^°S ^^" q + c = log sin 7 + log a = log (a sin q) ;
dp . 1 ^7 / .
.-. — — = a sin o ; also -— = a sin » ;
dt ^ dt ^
.'. \/] - e^ sin-0 - \/l - e- sin^0 = a sin ((p - 0) ;
and v/l - e^ sin^0 + \/l - e^ sirrO = a . sin (0 + 0).
Cor. 1. The constants a and a have a mutual depend-
ence;
dp dq , . , .
for •.' — . — = - e~ sm p cos p = aa sin jo . cos p ;
d^ dt
.'. aa = - e~.
CoR. 2. The preceding equation may be put under a
simple form ;
dp a sin q , // , .
for ••• — = -■ . ; ••• a cos a = a cos « + a (1).
dq a smp
But a, a, a" are reducible to one constant ; for if fj. be
the value of (p when 6 = 0,
- 1 + vi - e^ sm^fx , 1 + \/l - e^ sin^/x
a = ; » a = : ,
sin fi sin /u
a = a cos o - a cos p = (a - a) cos ^ = ; •
^ sin Hi
Substituting in (l) for a, a', a", we have
cos (0 - 0) { - 1 - \/l - e- sin^ fx]
= cos (0 + 0) {1 4- \/l - e-sin^/x} - 2 cos /m;
.-. cos (0 - 0) + cos (0 + d)
+ {cos (0 + 0) - cos (0 - 6) I \/l - e^ sin> = 2 cos m ;
or cos . cos - sin . sin v I - e^ sin^/x = cos fi.
39 *' DIFFERENTIAL EQUATION^.
CoK. 3. If f , -.-T =f(4>),
d9
•■• /(0) +/(^) = a constant = ^.
Butif0=iui, = 0, and/(0) = O; /. j3=/(m);
•••/(0)+/(^)=/(m).
Integration of Differential Equations of the first Order
and of the n^^ Degree.
be the equation ; P, Q, &c. and f7, being rational functions
of CG and y.
Let the equation be solved with regard to — ^ ; and let
dx
dy
X-i, X^, ^3, &c. be the values of - - , or j> thus found ;
then each of the equations p = X^^ p = X., p = X3, &c. when
integrated will satisfy the proposed equation, as also will the
equation formed of the product of all these integrals.
Since the diiferential equation arises from eliminating a
single constant*, raised to the n^^ power, from the primitive
equation ; and since each simple integral introduces a constant,
the solution will contain n constants, and therefore be more
general than that from which it is derived. But if we consi-
der that the constants are arbitrary, we may make each, equal
to the constant belonging to the primitive equation, and then
the result will be of the required form,
Ex.]. Let-^ = a^ .-. -^ = a, and -~ = - a ;
dcc~ dx da;
.-. y - am + Cy and y = — ax + c\
either of which satisfies the equation. Also their product
(y - ax - c) (y + ax - c) =
will satisfy it.
" For suppose y- cj: + 6'' = ; .-. p = c; .-. y - pv + p^ = 0, an equation of
ilic Hrst order and of the second dejujee.
DIFFERENTIAL EQUATIONS. 395
For differentiating we obtain
and making successively y = a,v + e, and y = - aa; + c\ we
. . dy dy .
get the results — = a ; — = — a ; as we ought.
dx dx
dy
Again from the original equation, since — = ± o ; .•.
dx
y — c = ^ ax, and squaring both sides, {y — c)" = a^x^.
This equation gives two lines, inclined at different di-
rections to the axis of x^ but both cutting the axis of y in
the same point ; and by giving to (c) different values, we
may have groups of such lines in pairs. And the integral
iy - ax ->r c) (y + ax - c) gives the same result, except that
each factor represents only lines inclined in the same direction ;
but by giving to c and c all possible values, and taking care
to collect together those straight lines in which c and c are
equal, we shall find the solutions comprised in the equation
{y - cf = a^x^, which is limited to the single constant c.
^ ^ dy^ /—
Ex. 2. Let — 1 = ax, or » = ± V ax ;
dx^
dy /- — dy , —
.*. —— = v ax, and — = - \/ ax ;
dx dx
,-. y = - s/ax^ + c, and y - — s/ ax^- + c',
4
each of which is comprised in {y - c)'^ = -ax^.
Ex. 3. Find the curve when s = ax •\- by.
Here
ds / dir dy
dx dx- dx
A 1 dy • 1 • ^ . dy
And •.• — IS obviously constant, let — ^ = m ;
dx dx
.'. y = mx + c, the equation to a straight line ;
Vi +
and V 1 + ] =a + h
X
Ex. 4. p^y + 2px = y ; y^ =■ 2ax + w^
396 DIFFERENTIAL EQUATIONS.
113. When the equation only involves x and p, and
can be solved with regard to a?, we proceed thus :
Since x =f{p) = P, and -^ = p;
da;
.: y = pw- fpa; = pP- fpP;
whence y is a function of p, and therefore of x.
Ex. I. Let X + ap = b\/l + p^;
.-. y = - ap^ + bp \/l +p^ - fp(- ap + l> V^l + P^)
ap^ bp / b . ^ y r,^
The elimination of p will give y in terms of x.
1 /r^
Ex.2. Let (l + p~)x=:l ; .-.x = -„ , andp = V ;
I +p^ X
.-. y = px - i ^ = pic - tan~^ p + C
= \/ X — X- - tan ~ 'S/ + C
dy ^ / dy^
Ex. 3. Let J7 -- = V 1 + T^;
a■* x^ p^ X p
X
; v/r
- ap
a?2 a" o? - x""
''' ^ " y/a- - x' ~ x/'d'^o^ ^ ~ y/'of^^
= - y/ d^ - or ; .*. y"^ + .r^ = a^,
which is the solution of the problem : " Find the curve in
which each of the perpendiculars drawn from a given point
upon the tangent is equal to a given line."
Ex. 2. Let y = px + - (l -i- />^) ; .♦. y = i-a (a + x).
DIFFERENTIAL EQUATIONS. 399
Ex. 3. y = p.v + a y/l + p^ ; y = c,v + a \/\ +
Ex. 4. Let {y - pw)'" = a""-' (- - .v] ;
... iiy^i^v'-^.,
\mj Km - 1/
this is the curve in which JD'" = o™ ^AT.
Ex. 5. Find the equation to the curve, when the rectangle
contained by the perpendiculars on the tangent, one from the
origin and the other from a point in the axis of .r, at a distance
2c from the origin shall equal a given quantity bi~.
y-px y+(2c-x)p
Here -y x 7==l — = ^ '
x/l+p' \/l + p'
whence if b^ + c^ - a^; y^ = ~ [a^ - {c - ocf] .
Prob. Find the equation to the curve PQR which cuts
any number of ellipses APB^ AQB, kc. described upon a
common major-axis AB, so that the areas APN, AQM, &c.
shall be of constant magnitude, PN^ QM being ordinates.
Let AN = X, NP = y, AB = 2 a, m" the given area, b the
axis minor of one of the ellipses, but which varies as we pass
from one ellipse to the other ;
b . ^ 1 ^, ,_
•*• y - -\/2ax - w^\ :. m^ = - Jxby/2aa! - c^?^
Now differentiating with regard both to b and x,
^ / 2 1 ^^ r /^ 2 r.
- 'sy^ax-iv'^ + -.-— Jx\/2ax - x^ = 0;
a a dx
jj) \/2ax -x^= 6£\/2«
.'. since jj) Y 2ax — X = oj^\/~a'X — x=: ma :
for ^ refers only to a particular curve, for which .*. b is con-
stant ;
b / „ db m^
.'. - \/2ax - .r + -T— .-— = 0,
a dx b
/ ^ db m'a
or \/2ax - x~ + — . =0;
dx o
.-. /, ^2ax-x' -—-=C,
b
. , 5 ring's/ 2ax - x^
f, \/2ax - x^ = C
400 DIFFERENTIAL EQUATIONS.
Now when x=0, ^^yj-lax - a?^= 0, it being a circular area;
also since the area APM is of constant magnitude, y must be
infinitely great when oc is indefinitely small ; therefore since
- 7w^\/2 ax - a^ = y y. C, C must = 0, and hence the equation
to the required curve is
rnv^aw — c^
y
\x\/^ax — aP
Integration of Differential Equations of the second
and higher Orders.
116. The integration of differential equations of the higher
orders is effected only in a few instances. We shall begin with
the most simple.
d" V
To integrate — ^ = X^ X being a function of x.
d?y ^ _ d (dy^
0). Letf^ = ^; .-^m-^;
'^ ^ dx^ dx \dxl
.■■f-I.X; .: y = f.!.X.
ax
and so on; the constants have been omitted.
d^y
Next to integrate -7-5 = i-
ax
dy d^y dp _dpdy _^ dp
.,^^=r, and^^C + /,F.
dy 2
d'v , dy d^y
Ex... Let--f3 = -';-ake-=p, ^ = ?;
dx 4 dx
r> x^ cx^ , „
'^ 4.5 4.5.6 2
DIFFERENTIAL EQUATIONS. 401
d'y 1
Ex. 2. Integrate .,
dx- ^ay
dp dp
••• -1- = P
d.v dy \/ay
by substitution ;
,. ?^ = 2^/^^, = 2^+^'/^
« y/a
dx \/a
"^y V \/y + y/b '
which may be integrated by making y/y + \/h = «.
hr ' • . ^P
117- To integrate equations involvings and q, put — ■
for q, and the equations will be transformed to those of the
first order.
Ex. ]. Find the curve in wliich the radius of curvature is
inversely as the abscissa.
'dy'^
\d.v^j a~
Here ,., = — ;
cl-y 9..V
~ dw^
dp
dy d'y dp dx 2x
dx ' dai' d.v' (l+ju^)- rr
p ,v^ W - of'
\/l + f «' ~ «'
1 a^ ^a'- (b^ - a,y
•■• P = ~ ^ "^ {b' - ai'f ~ {b' - a^y~ '
b ' -x'
Ex. 2. Find the same when radius of curvature =
dp
dx (I
■'* (1 +ff ^ ~^''
p a a + ex
+ c
26 Ui)
402 DIFFERENTIAL EQUATIONS.
I A''^ - (a + cxy
p^ (a + cx)'
a + ex
\/ X' - (a + cxy
Ex. 3. ^^ = \/i + -^^; a? = logJr(://-c,)+ v/'(2/-c,)'-l|.
Ex. 4. Intem-ate ~-., = g- + m — ^.
dy dp ^,
Let --^=jo; .•.— = §•+ wii>-;
d.t' d.v
dx 1 , dij p
and
dp ^ + »rap'' d/> j^' + mp-^
which are intcgrable, and p may be eliminated.
f^~y ^r dy^ dp
Ex. 5. Integrate — , = F+ fn f~7.i ^^ :j~ = ' +>»/>-
dp dp dp
But •.• -^ = p-r^; .-. p-^= F+ W2p^;
d.t? d?/ dy
d%
whence if p~ = 22r ; ^mz = F;
d?/
a linear equation of the first order and of the first degree.
This equation is used to find the velocity of a body moving
along a circular arc in a resisting medium.
118. To integrate the equation
d' y ^dy
dx- dx
dy r dry (du \
Make y = c^^" ; /. / = weA" ; -i^ = — + v' e'-" ;
^ dx dx" \dx J
r i du]
.'. ef-'^lu + Pu + Q+ — > = ;
du
whence u' + Pti + Q + —- = 0,
dx
an equation of the first degree and order ; but which is seldom
intcgrable when P and Q are functions of x. It however is,
when P and Q are constant ; lot /' = ^ ; Q = B;
DIFFERENTIAL EQUyVTIONS. 403
du ^ . --
.-. — f 7/" + J w + // = ;
dop
or — + {u - a) {u - b) = ;
dx
which is satisfied by making ii = a and u = b;
.-. y = e/"," = fi"^+'" = Cje"",
and y = e//' =«""+'■ =0,/';
either of these values substituted for y will satisfy the con-
ditions of the differential equation ; but the complete solution,
which must comprise two constants, is y = Cye"'^ + c.e' ; which
by substitution we find also satisfies it.
CoR. 1. If the roots oi u^ + Au + B = ohQ impossible,
a = a + j3 \/ - 1, and b = a - (i \/ - 1 ;
= e"^ 5 (c, + c,) cos l^x + (e, - c.) \/- 1 sin /3.r? ( .
Make c^ + c.,= A sin ^, (c, - c^) v - l = ^ cos ^ ;
.'. y = Je"^' {sin ^ cos j3.r + cos ^ sin (ix\ = Je"'^ sin {(ix + ^).
CoR. 2. Let the roots be equal ; or a = 6.
Then y = e'^''{ci + Co) = c^e"^ which has but one constant.
To find the second constant.
Suppose h = a + h; ■• y = c^e"'' + c^e'"''^^"'
= e"' {c, + Coe'"} = e"' {c, + c, + c.A.r + ^^- + &c.} ;
make c, + c. = c', c, h = c", and /* = ;
119. The equation -4, + P— + Q?/ = 0,
Ot tV CI iV
is seldom integrable when P and Q are functions of x ; it can
however be solved when P = ^;-^; and Q = ^-^Th^.-
For make a + bx = e^- ;
dz 1 dy dy dz _ d?/ \
(/,i; ~ rt + 6a? ' * dx d;? t/* dx a + bx '
dd2
404 DIFFERENTIAL EQUATIONS.
d^y cty d% l dy b
dw^ dz^ dec a + hx dz (a + 6.r)^
\dz' dz] (fl + h,vy '
whence by substitution, and multiplying by (a + />.r)%
dz dz
which may be integrated by the preceding metiiods.
] 20. To integrate the general equation
d^y d"~^y d"~^y
d.tf dx" ' dx" "
where _4, B, C, &c. L, are constant,
dy d^y „ ,„„ ^
Let w = e'"'; .-. — - = w?e"" ; -- „ = m e""-, &c.
t/,y dx
.-. m" + Anf~^ + Bm"'- + Cm"~'^ + &c. + L = 0.
Let a, />, c, &c. be the roots of this equation ; then
y = e''% 2/ = A 2/ = e"; &c.
will be particular integrals of the general equation, and the
substitution of each in it will satisfy it. Hence the complete
integral will be, by the introduction of n constants,
y = cie"^ + cj' + c.e'"' + &c.
Cor. 1. Should any of the roots be equal, as a = h ;
then for 016""^+ c 6*% put e"'(ci + c^x) ;
.-. y = e"^(ci + c.,x) + c^e'' + &c.
And if three roots be equal, and a be the equal root, put
e"'(ci + CoX + c^x^) for Cie'" + c.e'' + f^e''',
and so on for any number of equal roots.
CoR. 2. If pairs of roots be impossible, substitute for the
impossible exponential functions, the cosines and sines of the
circular arcs, to which they are equivalent.
Ex. 1. — -, + 7ru = 0.
dO'
,, du ,, d^u ., „
Let u = e'"" ; .-. ~ = me'"'^ ; . ,; = m~e"'* ;
dO dO-
DIFFERENTIAL EQUATIONS. 405
.-. m^e"'^ + 7i^e"'^ = , .-. m^ + n' = 0, and m = ± »/. \/- l ;
= (e'+ c") cos nQ + (c - c") y/ - 1 sin nO
= A cos (nO -^ B).
If c'+ c" = A cos ^, and {c — c") \/- l = - ^ sin li.
Ex. 2. — -^ + n'^?* + a~ = 0.
Make a" = w"/3, and 7^ + /3 = ?^; ;
d-7t7
.'. — — + n~iv = ; .-. u = - (5 -V A cos {716 + B).
Make 5 = e'"'; ,*. m''^ + 'ikm + f = ;
.-. m = - A; ± \/^ a// - A;' = - /<; ± a \/- 1 ;
.-. 5 = e-'"(cV^^ + c"e-'''^~) = Ae-^'cos {at + B).
Examples (l) and (2) are useful in Physical Astronomy;
Ex. (3) gives the space a function of the time, when a body
moves through the arc of a cycloid, the resistance varying as
the velocity.
_ d;^y d-y dy
Ex.4. ^-6-4 + 11/ -62/ = 0.
d,v a.r doc
Let y = e""' ; .*. m"' - Qm^ + 1 1 m - 6 = ;
.'. y = Cie' + c.e'" + CaC^".
Ex. 5. Let -4 - 3 T-^ + ^ 3^ - 2/ = 0-
cfcf^ dx~ div
.-. ?/ = e"^^ (cj + c^A' + c-^x"^.
Ex.6. — ? + 8 ^+16« = 0; .-. y =e-'' {r.-^c.x).
dw dx
„ d'y dy > ^,. / r. ^
Ex. 7. -r4, - 6 —^ + 34« = ; .-. y = ^e^" cos (5 + ruv).
dx^ dx
d?y \ dy y
Ex. 8. — ^ f + ^ = 0. Let .r = e- ;
.-. y = e' (c, + c,«) = X (r, + c, log x).
406 DIFFERENTIAL EQUATIONS.
d' 11 1 dv 11
aw a,' dw .v
.r
Ex. 10. Integrate — ^ - a^y = 0.
.-. y = ce'"" + 6*26 ""■' + Jl COS {B + r/.r).
Ex. 11. Integrate -— -v a y = i).
r,i l±V^ri _i±a/-1
Here wi^ + «* = O; — = r-— , and =
— / ax \ _±1 / ax \
.-. y = A e^/2 cos I fi + — p: 1 + ^, e ^-' cos ( B^ + --^ 1 .
^ . d^ y \ d- y
Ex. 12. Integrate -,- = — ~ .
^ dx' a' dx'
.•. ?/ = c, e' + 6ve " + c-j + 6*4 X.
d"y
Ex. 13. Integrate V = ^'
dx"
Make y = e""^; .-. r«" -1=0,
let 1, «!, a., as, 04, &c. a„_„ be the roots of this equation ;
.-. y = de"-' + c^e"!"'' + 036"-'' -^ &c. + 6'„e""-i'^.
121. To solve the equation,
cPw ^dy
^ + P^+Qy:=B (1).
dx~ dx
We shall shew that the solution of this equation may be
made to depend upon that of the equation,
~4 + P/+Q.y = (2).
dx' dx
To effect this, we proceed to apply to this equation, a method
called by Lagrange, " The Variation of the Parameters ;"
which consists in this, that if «/ = c'Vi + c'Vi be the solution of
the equation (2), we may assume it to be that of equation (1),
if c and r" be considered functions of x.
DIFFERENTIAL EQUATIONS. 40?
Let .*. y = c y^ + e'y., be the solution of (1) ;
dy ,dy, ,,dy2 dc dc'
•'• T = ^ 'f' + ^ ^ + y^ :r + y^^i~ ■
dx da: doa dx d,v
But as we have made but one supposition to determine c' and
c", we may make another ; let therefore
dc dc" dy ,dyi „dyo
Vi -1— + Vi -J- = ; .•.-— = c -— + c ~~ ;
dx dx dx dx dx
d-y ,d^~yi „^^y-i dc dyx dc" dy^
dx' dx' dx'^ dx dx dx dx '
whence by substitution in the original equation (l),
\d{tr dx I \dx dx
dc dy^ dc" dy.,
dx dx dx dx
which by means of equation (2) is reduced to
dc dy^ dc' dy^,
dx
H^
r dx
= R
dc
■'
2/i
dc
or
■ dx ~
y-i
dx'
dc
dx
(dyr
[dx
yi
y2
dyA
dx)
= i?
dc t
whence — '^ is found to be a function of x, and c = X^ + C,
dx
also similarly c" = Xg + Cg ;
••• y = C,y^ + C^yo +y,X^+ y.,X.,.
A similar proof applies to equations of a higher order.
d^y . n
Ex. 1. Integrate — ^ + a^y = cos px.
dx
d' y
The solution of the equation -r-+cL'y = is
dx~
y = c cos ax + c" sin ax ;
let this be the solution of the proposed equation ;
dy , . „ dc dc' .
,♦. -^ =i - c a sin ax + c a cos aX f 7- cos ax + ~~- sin ax
dx dx dx
= - c'asin ax + c"a cos a.i.
408 DIFFERENTIAL EQUATIONS.
de dc .
Since — - cos a. I' H sm aw = :
dx dx
ax - c a sin ax -
dc , dc
I
dc' . dc"
• . r-^ c a cos ax - c a sw ax - a -— sm acc + a~z— con ax
dx dx dx
a~y - a — - sm ax + a — cos ax :
dx dx
_ , sinoti + a- — cos ar = cos /3v dx
dc' ( . cos^ ax\ _
.•. - a -— sin oA' + -; = cos iix ;
dx \ sm ax )
dc \ 1
• ■• -7- = cos/3^sina.r = {sin(a + /3)cr + sin(a-/3).i'},
uiX a 2a
J rfc" 1 ^ T
and -— = -cosHa;cosQ,r = — }cos(a + B)x + cos(a - /3).ih
dx a 2a' V /-/ ( ^
.*. C = Cj +
1
1 J cos (a + /3) X cos (a - /3) .rl
2al ^TJ ■*■ a-(i r
{sin (a + /3) A' sin (a - /3) ^rl
2
1 /cos/3v J^ X sm am
sin a.x" . ,^
+ j^, -^ cos aoc.
I
Ex. 3. Integrate ^ +A^ + By=X; X =f(.v).
dx~ ax
Let a and 6 be the roots of the equation m + Am h B = 0:
.'. let 2/ =-- 6"'+ cV be the solution of the equation ;
.-. — = ace"' + 6cV^ + e'"-— + e' -r-.
a.f ax ax
.*. -— = acV^ + hc'p!"'.
ax
dx'' ax ax
.-. c'e"'(a' -r Aa + B) + c'V" (/r + Ah + 5)
rfA' dx
And a^ + ^icr, + 5 = ; Ir + Ab + B = 0;
rfc' , dc"
But be"' - + be''-- = 0;
dx dx
.-. (a - b) e'" ^ = ^, and - (a - b) e'' -^^-^'■>
dx dx
^ l,Xe-"\ c" = c, - ^ f^Xe-"' ;
a- b a - b
ax bx
a - b' a. - b'
410 DIFFERENTIAL EQUATIONS.
d'y d y
Ex. 4. Integrate ~ - 5 ~ + 6y = x.
^ dx' dx ^
.: y = c,&'' + c^e^' + _ ( .^ + _ j
SIMULTANEOUS DIFFERENTIAL EQUATIONS.
122. In the applications of the Differential Calculus to
physical problems, mutually dependent equations are fre-
quently found in which n + \ variables are involved, and n
equations are given : as most commonly the unknown quan-
tities are x, y, and t ; x and y being functions of t ; we sliall
first solve the system of equations which involve these quan-
tities. The method of solution is due to D'Alembert.
Let A- +B^+Cx + Dy = e,
dt dt ^
d V du
and Ji — +B,-^ + C,x + D,y = 0,,
dt dt
be the two equations : J, 5, &c. being constant, d and ^i
... dx dy
functions of t. By the successive elimination 01 — and —
dt dt
these may be reduced to the form,
dx dv
-^ + ax + by = T (1), -£ + a,x + b,y = T, (2).
dt dt
Now multiply (2) by m and add the product to (l);
.-. — ix + m%j) + (« + ma^ (x + ^ y) = T + m T^.
dt ^^ ^ a + mai
b + mb, , , 11 1 n
Let = m: and let m,, m., be the two values or m
a + ma^
resulting from the equation; also let a + mir7i = ri; o-\-7nMi=r.r,
.'. we shall have the two hnear equations of the first order,
— (.r + w,«) + r, (x + miy) = T -^ yn^T^;
dt
d
— {x + m.,y) + ?\, {x + m.^y) = T + /»._. 7', ;
.-. X + m,y = e-V {/,e''' {T + m, T,) + C{ ,
X + m,y - ir''^ )/ «'-' ( T + ni, 7\) + C, \ .
DIFFERENTIAL EQUATIONS. 411
dx , dy .,,
Ex. Let -^ +4.y + 5.v = e'; -f; + a? + 2?/ = C"' ;
dt "f
.-. — (.r + my) + (4 + 2m)y + (5 + m) x = e^ + we'^'.
dt
4 + 2w
Let = m; .-. m = 1 or - 4, 5 + m = 6 or I ;
5 4- Wi
••• * + 2/ = e-'' \f,e''(e' + O + C} = f e' + Je^' + Ce-«',
or .r - 4^ = e-'{ jje' (e' + e^') + C^j = Je' + Je^' + C,e-';
which give a? and y in terms of t.
123. Next to integrate the simultaneous equations,
^+{Ax + By + Cz)=^T (1),
dt
^ + (J,r + B,y + C,z) = T, (2),
-£ + (^3^ + B,y + C,^) = T, (3) :
where yi, B, C, &c. are constant, and T, 7",, T'g functions of t :
multiply (2) by m and (3) by m' and add ;
, B + mB^+ m'Bo , C + mCi + m'C,
where B = — : — , ? C = . , r »
A + A^m + A^m. A + m A^ + m A.^
U= T + T,m+ T^,m %
.•.'\iB' = m\ C'=m'; A + A]m + Aom'= M; ,v + my ■^mis = v;
dv
— + Mv = U, a linear equation,
dt
which integrated will give the relation between v and t : also
since from B' = w, and C = m, two cubic equations will
arise if m^, m^, ^3 ; m^', w/, ^3' be their roots, and if (7i,
fJo, CTg be the values of the right-hand side of the equation
when integrated ;
.-. .V + m^^y + mi'x = t7„
X + m.^y + m.'x = U^-,
,v + fn.^y + m-iZ = U-r
412 DIFFERENTIAL EQUATIONS.
124. To integrate the simultaneous equation of the second
order.
^ + aA' + 6y +r = () (1),
-^+ ajcr + fc.T/ + c, =0 (2);
ar
multiply (2) by m and add
d'
— - (a? + 7w V + c) + (a + maA { x -f y + > = 0.
\ a + ma^ a + maj
b + h^m
Make m = ; u = os + my + c\ a + a.m =^ - n \
a + m«,
du
.-. — - -n'-u=0\ .'. u = Ce"* + CjC" "' ;
dt
therefore if m^ and yn^ be the two values of m,
(a + «!?«,) (a; + m,,y) + c + w7,Ci = (a + m^a^ {c,e"' + c..e~"'},
(rt + «! Wo) (a? + /w^y) + c + w.Cj = (a + maaj) {c/e"' + c/e""'} .
Ex. _ = 3. + 4,-3: ^^-=8j,-..-5;
.*. a? = y + 4c,e^' + 4c2e-^' - SCjVV? _ Co'e^?^
?/ = § + c,e-' + Coe-^' - c/eV7 _ c/g-^/^.
Differential Equations containing more than two Variables.
125. Equations of this description, of the first order,
which, we in general shall suppose, involve the three variables
«r, y, 5r, may be divided into (l) Total Differential Equations,
(2) Partial Differential Equations.
Total Differential Equations.
Let rf ?/ = Pd.r + Qdy + i?rf.r be the equation, which may
be supposed to arise from the differentiation of
^ , du dii die
u = f'Lvyz); whence P = — ; Q=-— ,• J{ = — :
-f^ ^ ^ da; dy dz
and since when this is the case,
dP dQ dP dR flQ dR
- — = ; = ; and — = — r- :
dy d.i dz d.v dz dy
DIFFERENTIAL EQUATIONS. 413
we can always ascertain when an equation is a total difl'e-
rential.
Ex. Let du = — ^^ civ + dy + dz.
a — z a — z \(^ ~ ^y
dP I di^ dP y dR
Here -— = — = -7— , 3— = , ^2 = 7~ '->
dy a-z d.v dz {a - z) dx
diV a-z a-z
du , dri ,
.-. dy + --~dz
dy dz
= ^-^dy+ ~^^^ dz + ^f{yz)dy + ^f(yz) dz;
a - z "^ {a - zy dy dz
,. ^f(yz)dy + —f(yz)dz = 0;
•'.fiyz) = C;
... u = -^ + C.
a-z
126. Next to integrate the equation Pdx + Qdy + Rdz
P Q
= 0, which may be put under the form dz = - ^dx--dy;
P Q
or which, by making p = - —, 9 - ~ n->
may be written dz = pdx + qdy.
Now if this equation can be expressed by an equation
^ =/(.r, y, c), or f{.v, y, z) = c,
,, , , d(p) d(q)
we ought to have — - — = ,
'^ dy dx
dp dp dz dq dq dz
or - — I- -7— • -r - ~i — *" T~ * j~ '
dy dz dy diV dz dx
dz , dz
or ••• ■^ = q, and — = p, wc have
dy dx
dp dq dp dq
dy dx dz dz
an equation of condition, by which we can ascertain whether
414 DIFFERENTIAL EQUATIONS.
the proposed equation admits of the sohition /(.r, y, z) = c.
If we restore the values of p and q the equation becomes
U;^ dy)^^\dw dz) \dy dx) '^
when this equation or the preceding one holds, one of the
variables must be considered as constant, and the remaining
part of the equation integrated according to the rules given
for the integration of functions of two variables.
Ex. 1. {y + z) div + {x + y)dz+{z-^ x) dy = 0.
dP dP ^ dQ dQ ^
Here P = y + z, --=!=-- , Q:= x + z, -y = ~ = 1 ;
•^ dy dz dx dz
dR dR . ,. . ,• /> 1
R = X + y; — = 1 = ; .-. equation (2) is satished ;
dx dy
dx dy
.-. makinop dz = 0; + = ;
X + z y + z
.: log (x +z) + log {y + z) = cti {z) = log (Z) ;
.-. {x + z).{y + z) = Z;
.'. (y + z) dx + {x + z)dy + (x +y + 2z) dx = dZ ;
.-. — - = 2z; .-. Z = z'^ + C ; .'. xy + xz + yz = C
dz
Ex. 2. {ay - bz) dx + {cz ~ ax) dy + {hx - cy) dz = 0.
Make z constant ; .'. dz = 0, then
ad.v , ady ^. . i„g W - M , ,„„ z ,
CZ - ax ay - bz \cz - ax.
ady a(fiy - bz) dx a (bx - cy)
• ^ :: 1 --^ ~— + dz = aZi;
' ' cz - ax {cz — ax)" {cz — ax)'~
.-. {cz - ax) dy + {ay - bz) dx + {bx - cy) dz =
= \{cz - ax)]"dZ ;
.-. f/Z = 0; .-. Z = C; .-. {ay - bz) = C'{cz - ax).
127. If the equation Pdx + Qdy + Rdz = is not a
complete differential, but may be rendered so by the means
of a factor F, the equation (2) must still be satisfied : for,
nuiltij)lying hy F,
DIFFERENTIAL EQUATIONS. 415
is an exact differential ;
d.FP d.FQ _d. FR
dz dz dy
dF dF _
' dy dx
0,
FPdx +
FQdy +
Flidz
d.FP
d.FQ
d.Fl
dy
dx
dx
\dy
_ dQ^
dx J
\dx
dP
dz,
n^
dR
~ d^,
dP-
dz)
+ R
dF
' dx '
-P
dF
' dz
dR\
+ Q
dF
■ dz '
- R
dF
'Jy
Multiply the first of these equations by R, the second by Q,
and the third by P, and add : we have
/rfP dQ\ (dR dP\ (dQ dR
^Vdy- 'dxj ^^^'[d^-J^]^^- Vdz ' dy,
the same equation as in the preceding article.
128. When the differentials dx, dy, dz exceed the first
degree, it must be solved with respect to dz ; and can then
only be integrated when the factors of the equation so solved
are of the form dz - pdx - qdy - 0.
Partial Differential Equations.
129. It is here required to find %=f{xy) from one of the
partial differential coefficients, or from some relation existing
between them.
To integrate — = P; P being a function of x, y, z : we
dx
first integrate it on the supposition that y is constant, and
instead of adding an arbitrary constant after the integration
we add cj) (y) : similarly we add cp (x), if the equation be
dz p
.z = x''(p(y).
= 0;
dy
Ex. 1.
dz
• x = ax + (p{y).
Ex. 2.
dz az
dx X '
-. log z== log x"(p(y); .-
Ex. 3.
dz y'- + z
dx x^ -r y
dz dx
r- ' ' ■ y'^ + z' x' + y"
416 DIFFERENTIAL EQUATIONS.
y{z - x)
y y ' ' y'
tan tan"' - = tail"' 0(?/) ; ^^^ ^=^(^y).
„ dz cV^ — y- z x^ + M-
Ex.4. _= -^^ -; ^= :^0(y).
dx x^ + if X X ^
130. To integrate the equation Pp + Qq = R, in whicli
P, Q, R, contain at once, a-, y, z ;
dz - qdy
'.' dz = ]}dx + qdy; .-. p= — ;
dx
.•. substituting, Pdz — Rdx = q. (Pdy — Qdx).
Here there are two cases: 1st, Pdz — Rdx may contain
only z and x, Pdy - Qdx only y and x; 2nd, either or both
of these factors may contain all the variables.
Case 1. Let F be the factor which will make Pdz - Rdx
a complete differential dM, and 7^, the factor which will make
Pdy- Qdx = dN;
q.F . qF
.'. dM = .dN, which cannot be integrated unless —
Fi F,
is a function of N, =(j)(N); whence dM=^(p\N)dN\
.-. M=(p{N).
_, dz - qdy
iiiX. 1. px + qy = nz ; also p = ;
dx
.•. xdz - nzdx = q{xdy - ydx) ;
and to integrate xdz - nzdx, and xdy - ydx, we must mul-
tiply the former by - , , the latter by — ;
X" X ^ \x
Ex.2, px + qy = 0; .'. z = (pi-\.
Ex. 3. qx -py = 0; .-. z = q> (^'^ + y'O-
Ex. 4. np + hq = c ; .-.»= — + - d) . (ay - bx).
Ex. 5. px - q = 0!^; .: z = '- -i- (p. (y + log x).
Cask 2. Next let the variables x, y, z be found in both of
the functions Pdy - Qdx, and Pdz - Rdx; we can no longer
DIFFERENTIAL EQUATIONS. 417
integrate them separately, since z cannot be considered con-
stant in the former, nor w in the latter.
Lagrange observed that if these equations were integrated
conjointly : and if we call the integral of the former N: and of
the latter iV/; so that N=a, and M= b, a and b being arbitrary
constants; then the complete integral will he M=(p(N). But
that this method may succeed, one of the equations must involve
two of the variables only ; and its integral will enable us to
eliminate one of the three variables a;, y, z, from the remaining
equation.
The truth of this proposition may be thus shewn.
Since the equations iV = a, and M = b, are derived from
Fdw - Qdy = 0, and Pdz - Rdoo = 0, the differentials of JV=a
and M= b will be satisfied by the values of dz, dy, deduced
from these latter equations : hence differentiating and putting
.. dM ,, dM . dM
dx ^ dy dz
M,.dx + Mydy + MAz = ; and N^dw + Nydy + N,dz = 0.
^^ dy Q dz R , . .
But — = — ; — = — ; ••. substitutmg
d.v P dx P ^
But from the equation M = (p {N) ;
.-. M,dx + Mydy + M,dz = (p' (N) {N,dx + N^dy + N,dz};
hence My.{Pdy - Qdx) + M, . {Pdz - Rdoc)
= 0' (N) {Ny (Pdy + Qdx) + N,(Pdz - Rd.v\;
.'. Pdz-Rdx=- Z'~^'t!:Zl iPdy-Qdx) = -u.(Pdy-Qdx);
M:,-M.(p{N)
^ R + w.Q, .
.-. dz=— — dx-(t).dy;
P ^
whence /> = , 9 = - &) : which substituted in the
original equation Pp + Qq = R satisfy it; and therefore the
assumption that M = cp (N) (which is derived from the in-
tegration of Pdy-Qda; = 0; and Pdz - Rdx=0) is the
solution of the problem, is completely justified.
27 ' Ee
418 DIFFERENTIAL EQUATIONS.
Ex. 1. poo^ - qxy + if = ;
.*. X'dz + ff^dx = q {ardy -\- yxdx) ;
.-. x'^dy + yxdx = 0(1) ; and x^dz + y^dx = (2):
n
from (1), xdy + ydx = ; .-. xy = a = N -^ .'. y = - :
from (2), x'^dz + — rfo? = ; :. z = 6 = il/,
or^-— = + (6-?/)7; .-.• =0
;^ - r ^ \z - cl
xy 2 . /?/
Ex. 9- px + qy = -^; z = xy + f { -
z ■ V''r
Ex. 10. z - px - qy = m {x +17 + 5?);
/ X" = ^ '
rf?/ d.r di^
1,2,
g + -TJ»+-w=0;
V V
whence, treating it as an equation of three variables and a
function of {ipxy%)^
1
dp = 0; .-. p = a; q = - '->
a
.'. dm - adx - -^ = ; .\ s; - ax - - = ((pa) ;
a a
y '
whence, by differentiation, —^-x^cp (a).
a
d%^ d%'
Ex. 2. Let — + — = 1 ; or p + (/- = 1 ;
- ax -ys/l - a'' = (f){a).
ay
x/i
^ and Q being
functions of ce and ?/.
d z dp
If -— = p ; .*. Pp = Q, a linear equation ;
d.v dcD
whence p = e" {^e~"Q + (j/) } where w = f^P ;
.-. sr = /,e" S/.e-"Q + 0y} +/(2/).
Ex. 1. Let..2/— = (n-l).y.- +«;
f{y)-
4g|2 DIFFERENTIAL EQUATIONS.
w.(w - l) {n - \)y
Ex. 2. a?« =607 — + «2/;
docdy ax
137. To integrate J?r + 6's + Tt = F, where 72, ^y, T
and F are functions of x^ y, %, p and q :
dz , d^% ^ d~z , J .
•.• « = — ; .-. dp = — - dx + -— -— rfy = rdx + «d?/,
^ dx dx^ dydx
and o = — ; .-. dq = —-^dy+ -— — dx = tdy + sdx ;
' dy dy dxdy
^^^dj p-sdy ^ ^^dq^sdx^ ...substituting
dx dy
Rdpdy + Tdqdx - Vdxdy = s{Rdy'' - Sdxdy + Tdx"^) ;
it is unnecessary to integrate the two members of this equation
separately ; for if we can integrate one of them so as to have
the integral N = a, and by combining this with the other arrive
at the integral M = />, M and N being functions of x, y, z, p
and q, we may prove, as in a preceding article, that M=(p{N):
and this result will give an equation with which we must
proceed, as with an equation of partial differences of the first
order.
d^z , d'^z
Ex. 1. To integrate ——k = c- . --„ ; or r = c"^ :
° dx dy
since dp = rdx + sdy ; dq = tdy + sdx ;
.-. dpdy - c~dqdx = s{dy" - c^dx^) ;
dy"^ 2 dy
.-. -^ = (• ; .•. — = =t c ; .'. y - ex = a: y + ex = ai :
dx dx
and dp . — - - (fdq = ; .-. edp - e'dq = ;
dx
.-. p-cq = b = (p'(a) = (p'iy - ex).
But p = ^ ~ ^ y ; .-. d« - (h'(y - ex) dx = q (dy + edx);
dx
.'. dy + crf.t = ; .•, y + ex = a, y = a - ex ;
DIFFERENTIAL EQUATIONS. 423
.'. dz - (p'(y - cx) dw = dz - (pi'{a - 2cx) dx = ;
.-. z-S ^ \ d'w ^
+ i? - — + &C. -— ; + &C.
\ dx / dx-
r,^ dP d'Q dR \
J^\ dx dx' dx^ )
428 CALCULUS OF VARIATIONS.
8. Thus the variation of ^ T consists of two distinct
parts, one of which is under the sign of integration, and the
other is not ; the latter is affected only by the variations of the
extreme values of y and w, the former is dependent upon all the
values between those extreme ones. Let x^, t/u ^^'a? V-z be the
values of x and y at these limits, then the total variation of j^V
= V,,^X^ - V,hx, + (Pa --T^ + J-i- ^C.)«^2
(t 01/2 ^ *^2
dxi dxi
The part under the sign of integration must be taken
between the same limits.
9. If ti =f{xyz) and x still be the independent variable,
then, since the differential of V, may be put
dV= Mdx + Ndy + Pdp + Qdq + &c.
+ N'dz + P'dp + Qdq + &c.;
.-. ^ F = M^x + Ndy + Pdp + Qhq + &c.
+ N'dz + P'dp + Q'^q + &c.
We shall have for the new variable z, a series of terms,
involving N\ P', Q', &c. similar to that in which JV, P, Q, &c.
are introduced; .'. if ^^ - p'^x = w\ the total variation will
be expressed by
' ^ dQ d'R
J/.F=F3.^(P--+^-&c.)»
, dQi dR' ^ ,
.-(Q-^.&c....)^'
dx dx
, dR' , dw ^
+ (Q' - -— + &c....)-T- + &c.
dx dx
dP d'Q
■' dx dx-
' dPy d^Qr . . '
CALCULUS OF VARIATIONS. 429
Maxima and Minima of Integral FormulcB.
10. We proceed to apply the results of the preceding
article to the solution of some geometrical problems, involving
the lengths and areas of curves, the surfaces and volumes of
solids; when these quantities are, within certain limits of the
variables, the greatest or least possible.
Now we know that if w, any function of x and ?/, be a
maximum or minimum, dw = 0; and by the same kind of
reasoning which has been used to establish this proposition,
it may be shewn that under the same circumstances the vari-
ation of u also vanishes; but if ii=Vdx, we have seen
that between the limits of a?j, t/i, x.>, y.^,
l.LV= V,^x,- V,lv, + iP,--^ + kc.)w,
•'' ax2
- (Pi - ^' + &c.) w, + &c.
.^^ dP d'Q d'R ^ ^
And since when ^Fis a maximum or minimum, ^^r = 0;
therefore the two parts of which the variation of /^ V is com-
posed must separately be put = ; one part will determine
the relations between the co-ordinates of the extreme values of
the required function, the other the function which possesses
the required maximum or minimum property.
dP d^Q , ,
11. Thus from iV ---+-—- &c. = 0, and the
dx dx^
equation dV = Mdx + Ndy + Pdp + kc. maybe found the
curve or function, which is the object of our enquiry, and
from V^Sx.^ - Vi^Xy + P^w^ - P,w,&c. = 0, the position of its
extreme points may be determined ; if however the extreme
points be fixed, Ix^ = 0, and Ix^ = 0, and the latter equation
disappears.
12. Thus, if the shortest distance between two given
points be required, the former equation will be quite sufficient
for the problem ; the constants of the integral being deter-
mined by the co-ordinates of the given points ; but if we wish
to find the shortest distance between two given curves the
latter equation is also necessary, since it determines the points
430
CALCULUS OF VARIATIONS.
in the two curves to which the
drawn.
Its use may be thus illustrated ; let PP,
given curves, and PQ, P, Qi two
curves drawn between them, and
let PiQi be derived from PQ by
writing x + Sx, y+Sy for .v and y :
shortest distance is to be
ah
dy ^ dy .
let — ^ = m ; and —-= n be
dx dx
ni Ki
y^
n ; (since the point P
the equations to PP^, and QQi ;
then if ■t'l^j, x.,y.> be the co-ordinates
of P and Q ; .'. V^ = ^ ; and -^
6x^ dx2
is always in PPi and Q in QQp) and between these equations
and that of the limits, F.^Sx^ - V^^x^ + &c. = 0; two of the
quantities, as ^^/i ^^^ ^V^ ™^y ^e eliminated, and the two
independent variations, ^x-^ and hx^ will be left, the coefficients
of which, being separately put = 0, will give equations, by
which and from the given equations to the curves, the points
P and Q, may be determined.
We have here tacitly assumed that Idy^ = and ^dx^ = 0:
if this be not the case, some new conditions must be fulfilled
by the limits, which will enable us to introduce the higher
difPerentials of the equations of the given curves : by means
of which some of the variations ^dx^ Sdy, &c. may be elimi-
nated, and the coefficient of the remaining variations separately
put = 0, and the co-ordinates of the extreme points and the
given conditions fulfilled.
13. We shall now deduce from the equation
&c. = 0,
_ dP d'Q
dx del
some formulae of great use in the solution of Problems of
maxima and minima.
Let dV= Mdx + Ndy + Pdp + Qdq + Rdr + &c.,
d'R
J.
(I) Let all but A^ and P= 0;
dP
, ^, dP d'Q
and N - -^- + ~—
dv dx^
+ &c. =0.
dV ,, Pdp
— - =Np+——.
dx dx
■.ml N-''~l
dx
N =
dx
CALCULUS OF VARIATIONS. 431
dV dp dp ^, „
.-. --=p-- + pJi. .-. V=Pp+c.
do) dv d.v
(2) Let all but 31, N, and P be = 0;
r.^-I=M + ~iPp + c); r. V = f^M+Pp + c;
d,v dx
(3) Let M =0y N = ; and all the terms after Q = ;
dx da? dx dx dx~ dx
dV dQ dq dp rr ..
dx dx dx dx
Cor, If 31 does not = ; F = Q7 + cp + c, + f^3I.
Prob. 1. Find the shortest distance between two given
points in the same plane.
f^v-rs/^.% = f.vT^-^
Here
dx'
P
.-. 31=0; N=0; P= ,~ — - . Q = 0.
v/i
+ P'
But N- '~ + kc. = 0; .-.—- = 0; .. P = C
dx dx 1 + p
c ,
.-. p = = a ; .-. y = ax + 0,
v/l - c"
the equation to a straight line ; the constants a, b may be
determined by the co-ordinates of the given points.
Prob. 2. Required the curve of quickest descent between
two given points.
Let y be vertical and be measured downwards.
ds
Then time = / -»;=^ = -;= / 7-^ = -7= / V;
Jx\/2gy \/2g'^s \/y V^g-^-
432
CALCULUS OF
VA
p.
RIATIONS.
V
2,1 '
\/yV^
^f
^/^
p + c; -
Vy
vy \/i + p^
+ 6;
1 , /2 a
= -7=- ; .-. v/i + p- = V — ;
\/y vl + p^ V 2 a
Ila - y . . , , .J
.'. p = \/ the equation to the cycloid.
y
Prob. 3. To find the shortest distance between two given
curves.
From Prob, (1) V=\/lJrp^\ p = c; y = ax + b the
equation to the line which is the least distance required.
^ ^y ^ dy 1 1 ■ .
Let — = m, and — ^ = n, be the equations to the two
dtV d.v
curves, and yi x^^ y.^ x.^ the co-ordinates of the points in which
the shortest line intersects them ; then since ^^i, ^x^ are the
variations of y^ and x^ as we pass from one point to another
• 1 ^y
adiacent one m the curve — = m ;
•' dx
hi dyi Sy., dy2
= =ni; and ^ = -— = w.
6X2 dXi 6X2 dx2
But r,^cj?2 - V^^Xi + P2W2 - PiW^= 0,
whence since the variations of the extreme points = 0,
V^^Xi + P,Wi = ; ¥2^X2 + P2W2 = ;
.-. Vjx, + PXSy,-p,Sx,) = Oil); ri,i^, + P,(^2/,-p.>,)=0(2);
r. 1 1
from (1) Fi + P^m - P^py = ; .-. m =p^ = = - - ;
"i P\ ^
V. 1 1
from (2) v.. + P,n - Pp., = 0; .\ n = p, ^= =--;
P- P2 c
.•, 1 + cm = ; and 1 + cw = ;
which shew that the line must cut both curves at right angles.
Also the equation to the line being y - y, -- -^ ^ (.r - x^) ;
X., - jy p" J
Prob. 6. Find the curve of quickest descent from one
curve to another curve, the velocity being that from a hori-
zontal line.
Here V = t^' • • n = V^f^-y ^ th^ equation to the
^/~y ' " y
cycloid ; the equations of the limits give
V^ Iv^ + P^^y, - P,p,^.r, = ; Vj,x., + Pjy, - P,pJ,v, = 0,
, . . . ^y\ T "^Va ,
from which since ^— = m, and r — = w, we have
Ki - P^p^+ P, w = ; and l\ - P.,p, + P.n = 0.
1 p p
But l\ - P,p, = c= ^y^-, P= - ^ = -^=;
.-. --= + ^ = 0; and . + ^ = 0;
y/2a V2a \/2a y/ 2n
,-. 1 + /),»? = ; and 1 + p..n = 0,
which equations shew that the cycloid cuts both the curves
v/.^'L" y ,
at right angles, and from p = S/ " — ^^ , we see that the
y
base of the cycloid coincides with the horizontal line from
which we have supposed the body to have commenced its
motion.
28 F^.
434 CALCULUS OF VAUIATIONS.
PiioB. 7. To find the curve of quickest descent from
one given curve to another given curve, the motion com-
mencing from the upper curve.
Let 2/, be the value of the ordinate at the point at which
the motion commences, y the value at the end of time t.
Then time = ^= [ ^ ^ ; .-. V = ^. — ■
V 2g J, y^y -y, \/y - y,
In this problem the function V involves y^ one of the vari-
able co-ordinates of the limits; in such a case we must add
rd V
the term ^f/i j to the equation of the limits, and then
the whole variation of the f^V will become
•' ■ Jr ax,
/(
dP d-Q
dcV
Now referring to the problem,
N- ^ + ^ - &c.)z 1 /—
Vy-yi vy-yiv^ + p- *-
the equation to a cycloid, the cusp being at the point from
which the motion commences.
. , rdV dV ,^ dP ^^ dP
Next, to find / -— ; -—= - iV = - -- , ••• N-~~ =0;
Jxdy^ dy^ dx ax
J^rfi/i ^,, ax
.-. Vjx., - V,^x, + P.MU - P,Wy -h (Pi - /\.)^2/, = 0,
or {V,- P.p.;) Sx, - (F, - P.MJ,) li\ + P.hh - /\
also '.• — = - ; -^ = - ; .'. X = - z + c\ y = -z +c ,
dz c d% c c c
the equations to the projections of a straight line.
Prob. 2. Find the equation to the shortest line that can
be drawn upon a given curve surface.
436
CALCULUS OF VARIATIONS.
Let dz = pd.v + qdy be the equation of the surface ; then
the variations of the co-ordinates ,%\ y, s: which are under the
sign /must satisfy the differential equation ; .-. ^X:=po.v + qly\
whence we have from the part under the sign of integration,
^ dw dz , , dy , d^
d.-— + p.d.-— = 0; andd.-^ + q.d. — =0;
ds ds ds ds
whence, having found p and q from the equation to the given
surface, the equations to the curve may be found.
Ex. 1. Let the surface be a surface of revolution;
.-. z = (p {.v^ + y') ;
.-. p = 2cr0'. {ar + y-); q =^y. = '>-
d^tV d'y
ds' d?
But if .?r + y- = r'\ and = cos *-;
yd- a: - a; d'y = d . (f-dO) ;
de rdO c
.-. nuegratmg, r^=c; ■■ -j^ = y
But — is the sine of the angle at which the shortest bne
ds ^
cuts the generating curve or meridian, hence if (p be this angle,
.6' c
sm cp = ~
r ^x^ + f
(ok. hmcc r -— = f,';
ds
CALCULUS OF VARIATIONS. 437
dy' "^ dr") '
dO" ds' / ,d0' dz'
•• r' -— = c- , ., = 6- 1 + /
dr- d r \
dO ^c /\ + 2 V 1 + p-
.-. F, = Pj9 + C ; .-.y-k-a \/\ + p' = /^ + c ;
V 1 + ;)■-
V 1 + jt)"^ 2/ - c
rfcV 2/ ~ <^ /
dy ^a' -{y-cf yy ) ^
.-. {x - c^y + (y - cy = aK The equation to the circle.
Prob. 2. Find the curve in which a chain of given
length I may hang, that its centre of gravity may be the
lowest possible.
^=def
>th
X
=
of centre of gravity
.-. V,
— i- + a\/l +p";
dx
■^ ^' dx 1 I ^^
1 ""^ "l^.
del'
.1? + ^ a? + ^
Zc " a ' "^
+ |3 + \/(<^^ + /3)^' -
p P^f' + lap
ly/i +p' '
\/l + p
a
fj — -
P
- = log
a
v'i^ + ^y-a'
a-]
— >, the catenary.
440 CALCULUS OF VAHIATIONS.
Prou. 3. Find the curve, wliich of all those that have
the same length and include the same area, shall by rotation
round the axis of cV generate the greatest solid.
Here ^F, = j.r{Try' + a \/\ +• p^ + by).
Prob. 4. Find the Brachystochrone when the length of
the curve is given ;
Prob. 5. Of all curves which include a given area, the
circle has the least perimeter.
Prob. 6. The curve which by revolution round its axis
generates the greatest solid under a given surface, is the circle.
Prob. 7. The length of curve being given, shew that it
will generate the least surface, when it is the catenary.
Prob. 8. If from a point two straight lines be drawn,
and their extremities be joined by a curve, so that the area
included is constant, the curve will be a circular arc, when
its leng-tli is a minimum.
T II E E N 1).
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S^.iK
1035
NOV ZA mi 6 V
k?H 22 1936
JUL 21 1937
:7 1 2
tUj -^.J Jy,^/
J^fTUR'^er TO
JUL 3 1 1972
'^£^B lo 1938
ciR.W)G2«'7Q
REC.
'^CT kn 'am
N0V21194QM
: >1'72 "5PMr> 5
x^z^t
mi 28 1978
LD 21-lOOm 8,'34
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865714
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