If the equation to the surface is

Au? + By + Cxe + D=0, the equation to the tangent plane is

Axx, + Byy, + Cxx, +D=0; and the equations to the normal are

Aw,
X - X= (x-%1), y-Y=

(x– x).
Cx
If the equation to the surface is

My + Nx+ Px=0, the equation to the tangent plane is

Myy + Nxxi + P(x+x)=0. If three planes perpendicular to each other are tangents to the surface

Byi
C%

the locus of their intersection is the sphere

x + y + x = a + b2 + co. If a conical surface circumscribes a surface of the second order, the line of contact is a plane curve.

If any number of planes passing through a given point intersect a surface of the second order, and at the lines of intersection conical surfaces be circumscribed, the locus of their vertices is a plane.

(H. A. G. 364, 85; G. G. A. Ch. xx; Biot, 335_8.)

m

DIFFERENTIAL CALCULUS.

a

dzU2 U2

(1.) Differentiation of algebraic functions.
dz(u, + u, + &c. + Un + const.)=dzu,+ dru, + &c. dx Un.
dau=ad, u, if a is independent of x.
dm = mcm-1

du, dzuju,=udzUQ + dzu,=U,U,

+ U1

dzu, dr U7 Ug... Un=UjU.... Un

+ &c. +

U1
vdzu – udzv udzu dl

o
ve
uu....um UUg... Um sduq

+ &c. + 010....Un 0.02...un U1

du,

-&c. 01

VM

dzUy

dua

+

B

dx

dau, +

dum

Um

dou expresses the same quantity which has usually been

du denoted by

(See L. C. D. Vol. II. p. 527.) dx d. (Sim 4m + const.) = S„,dąUm

-P

S
P, P,,

{s.

dzu,

1

(2.) Differentiation of exponential functions.

log, d, log x=

d, logax= dz€*=6".

dza*=log, a.ak. de sin x = cos x.

de cos x = dtan x=(sec x)?

d, cot x= - (cosec x)? de sec x=tan x . sec X.

d, cosec x = – cot x. cosec X. de vers x= sin x.

de covers x=

. sin .

-COS X.

a

- a

1

-1

a

devers

de covers (2ax — xo)

(2ax — ~?)}" Succi IVE DIFFERENTIATION. (3.) dzw" =nw" – 1. da" =n(n-1)–

21 2 &c.

&c. doc" =n(n-1)...(n-m+1).on – m. d) * = (m - 1)...2.3.1. d. Uv=udzv + vdzu. da uv=uda v + 2d,u.dv + vd; u. di uv=ud; v + 3d,u.div + 3d, u.d.v+vdu. &c. &c.

-
d.". Uv=du.v+ndon-tu.dv + d

1 2
nn
+

d
2

sin ,

B

(4.) d*a*=(log a)"a".
d” log x=(-1)"– 11.2.3...(n-1)x-n.
d, sin x=cOS X,

d, sin = - sin 8,
d; sin x=
COS X,

do sin x= &c. &c.

&c.

&c. din-1sin x=(-1)"- cos x, 0,2n – sin x=(-1)"-1 sin æ. d, tan x=1 + (tan x)", dtan x=2 tan x {1 + (tan x)?}, d; tan x=2{1+ (tan x)?}{1+3(tan x)}}. &c.

&c. de sec x=sec x. (sec æ) – 11. di sec x = sec X. {2(secx) * — 1}. di secæ=sec x. {2.3(sec x) – 1}. (sec xo) — 17. &c.

&c. These differential coefficients may be adapted to cosa, cot x, and cosec æ by substituting these quantities for sin X, tan a, and sec x respectively, and changing the signs of those coefficients the index of which is an odd number. (5) d, sin-la=(1- °)-%. d; sin - 1x = X(1 — «?)-*. d; sin - 4x=(1 — xo) - * + 3x2 (1 — «)'- . &c.

&c.

m

m

2n - 2

2n-1.a-1;

ß d,2m – 1:tan x
(-1)^ — " cos a

(-1)"-" sin a
S.

a-1+Sin 2m 2n +1

| 2m - 2n d. 2m —:tan x (-1)^-^ sin x

(-1)"-? :S

-21 – 2.a-1+S 2m - 2n

2 m - 2n
(-1)"–1 cos x

(-1)" sin x
, 027-1
2r 2

2r-1

m

-n-1 cosa

m-1

n

n

1

-1

d, tan–1x=(1 + ) -1.
do tan- X= — 2x(1 + **) -2.
di tan-1x= -2(1 + x^) -2 +28.x(1 + $0%) -3.

&c. = &c.
de sec-1x=x-1(202 - 1)-5.
di sec-1x= --%(— 1)-}– (— 1)-8.
di sec-1x=2x-3(no – 1) -*+=(22 – 1) - +30672 –

The coefficients of cos - lx, cot-la, cosec-1x may be found from these as above.

(L. D. C. 34-48.) (6.) d P(u)=d.P(u).du. (L. C. D. 11; L. D. C. 16.)

DIFFERENTIATION OF FUNCTIONS OF SUPERIOR ORDERS. (7.) Let d20(w)=''(x), then

d29"(x)=$'(x).$'$(x).'$*(x)... 'øn-(). $
d, sin”x=

= cos X.cos sin x .cos sinoX...cos sin” – 1X.

n

n - m

a

m

w d":P(u)=Smd na p(u)

{ar-1=d":4}. See Appendix. A few applications of this important theorem may here be added : dm:u" =ā".a”, {ap - i =d," – 1:0}. d":€" =€".Sm

{Ap-1=d":0}. d :sinu ton

m. a2m =S,(-1)m-1 cosu

+ Sm(-1)" sin u

n – m.a"

n

m

n'-2m

n - 2m +1

2 m

-1

2 m

d":cos u

n - 2m +1 alm

2m - 1

=Sm(-1)" sin u

+ Sm(-1)" cos u {am-1=dy:u; r= {n, if n is even, r=}(n— 1), if n is odd.} B d29"(x) = P.mp'$*-'(x). y desin” x=P mcos sin" – 18.

vP-1

n