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If the equation to the surface is

My+N+Px=0,

the equation to the tangent plane is

Myy1 + N≈≈1 + P(x+x1)=0.

If three planes perpendicular to each other are tangents to the surface

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the locus of their intersection is the sphere

x2 + y2+x2= a2 + b2 + c2.

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.)

DIFFERENTIAL CALCULUS.

(1.) Differentiation of algebraic functions.

d2 (u 1 + u2 + &c. +un+ const.)=d ̧u1+d ̧u2+ &c. dxun dau=adu, if a is independent of x.

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α

v

u

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du

+

სი.

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+ &c. +

du,

m

Um

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n

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d3 uv=ud3v+3d ̧u.d2v+3džu.d ̧v+vdžu.

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x

&c. B

d ̧ sec a = sec x. (sec ∞)o — 1]3.

d2 secx= secx. {2 (sec x)2 — 1}.

d2 secx = sec x. {2.3 (sec x)o — 1}. (sec ao) — 1 ]3.

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These differential coefficients may be adapted to cos x, cot æ, and coseca by substituting these quantities for sin x, tan x, and sec a respectively, and changing the signs of those coefficients the index of which is an odd number.

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1

-2

d3 tan-1x = − 2 (1 + x2) −2 + 23.x2 (1 +∞2) −3.

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The coefficients of cos-1x, cot-1x, cosec-1 may be found

from these as above.

(6.) d2p(u)=dup(u).du.

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DIFFERENTIATION OF FUNCTIONS OF SUPERIOR ORDERS.

(7.) Let d2p(x)='(x), then

n

d ̧p"(x)=p′(x).q′p(x)·p′p°(x).....p′p”-1(x). B

d, sin"x = cos x. cos sin a. cos sin3x.....

COS sinn

n-1

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A few applications of this important theorem may here be added:

d":um=w".a", {a,-1 = dr1:u}.

i

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{am-1=d":u; r=n, if n is even, r=(n-1), if n is odd.}

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