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 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. α v u du + სი. + &c. + du, m Um n 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. 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. 1 -2 d3 tan-1x = − 2 (1 + x2) −2 + 23.x2 (1 +∞2) −3. The coefficients of cos-1x, cot-1x, cosec-1 may be found from these as above. (6.) d2p(u)=dup(u).du. 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 A few applications of this important theorem may here be added: d":um=w".a", {a,-1 = dr1:u}. i {am-1=d":u; r=n, if n is even, r=(n-1), if n is odd.} |