· (d2x)2 + (d2y)2 + (d2≈)o\*. (L. D. C. 179—86; L. C. D. 351-60; L. Mec. Cel. Liv. 1. Ch. 2.) RECTIFICATION. (18.) Let s be the length of the arc of the plane curve, y=4(x), then ds=1+(d ̧y)o|*, and 8=ƒ,1+{d ̧p(x)}2\*. The value of s between the limits = a, and x=b, is (Lx=b−£x=a)1+ {d ̧p(x)}2]1.* (19.) Let u=4(0) be the equation of the curve referred to polar co-ordinates, then 2 d¡s=u2 + (d ̧u)®]3,, and s= {$(0)}2 + {d,$(0)} 3 \*. θ (20.) Let y=(x), ≈=√(x), be the equations of a curve of double curvature, then d ̧s=1+(d ̧y)2+(d ̧≈)°|13, and s = =√1+{d ̧p(x)}2 +{d ̧¥(x)}2]*. ́then 2(1+e)8=2 e11 − (1 − e3) (§,— €1.d ̧‚ 81) +281. * This notation will be explained in the Appendix. (1) By a similar transformation, we may obtain an equation of the same form between 8, and s,; from these equations we have 29 (2) we should have a similar equation between 8-9, 81, and s; and generally we may obtain a series of equations from which, any two consecutive values of s being known, any others in the series may be determined. By this process the rectification of any proposed ellipse may be made to depend on that of another ellipse of either much greater or much less eccentricity: in either case an approximation may be made by a series, as in (Art. 17.) (22.) The arc of an hyperbola may be thus referred to elliptic arcs: let 1 be the eccentricity, e the semi-axis major, and S the arc; then the equation of the hyperbola is Let x=e{1+(1—e°)(tan (p)°}, then y = (1 − eo) tan 4 ; (cos $)2 and _S=tan 4 { 1 — (e sin 4)°} - e2 + {1-(e sin ()o¦13 -- =tan {1-(e sin ()2}} ·s + (1 − e2) (s — e. By changing e, P, s, S, into e̟1, P1, 81, S1, and substituting for s11.de, its value obtained from (1), we obtain - =tan p,{1-(e, sin p1)°}+2e, sin 1 + §1 − 2 (1 + e1) s. S1 = (23.) S X {a+a‚x+a ̧x2+a ̧x3 +a ̧x1}§. which depends on the M M where Q={(1+p3u2)(1±q°u2)}', may be reduced to the form the value of the two first, and sometimes of the last, may be represented by elliptic arcs. (L. C. D. 502—11; Legendre, Exerc. du Calc. Int.) QUADRATURE. (24.) Let y=4(x) be the equation of a plane curve, and A the area contained between the axis of X, the curve, and any two ordinates (a), and (b), then d ̧A=4(x); and A=(=-=a) $(x). Sxy=xy-f,x=yp ̄1(y)−√,4 ̄1(y) ; this will frequently be found a convenient transformation, when the function is not readily integrable. (25.) Let u=4(0) be the polar equation of a curve, and the area contained between two radii and the curve, then The value of A between the limits = a, and 0=ẞ, is (26.) Let x=4(x,y) be the equation of a surface, and S the area of the surface intercepted between the planes xx, yx, and two planes respectively parallel to them, then and d ̧d ̧S= {1+ (d ̧≈)2 + (d ̧≈)2 ¦1, S=£x£v {1 + (d ̧*)2 + (d,x)o }§. If the surface is bounded by the plane xx, a plane parallel to yx, and the plane y=ux, then and d ̧d1S=x {1+ (d ̧*)2 + (d ̧≈)o¦1, S=Suf2x {1+ (d ̧≈)2 + (d ̧≈)°¦1. It is indifferent in what order the integration is performed, but the proposed limits must be assigned after each integration. CUBATURE. (27.) Let V be the volume contained between the three co-ordinate planes, two planes parallel to xx, and yx, and the portion of the surface intercepted between them; and let x=4(x,y) be the equation of the surface, then If the volume is contained between the co-ordinate planes xy, xx, a plane parallel to yx, the plane y=ux, and the portion of the surface = p(x,y) intercepted between them, then d1d ̧V=x.p(x,y), and V-fufx.(x,y). If the surface is generated by the revolution of a plane curve round the axis of x, then (28.) Let x and y be functions of t and u, the co-ordinates of another system, such that 1 2 then SSV becomes ftfu V (P1 Q2- P,P1), the values of ☛ and y in terms of t and u being substituted in V. Let x, y, and ≈, be given functions of a new system of co-ordinates t, u, v, such that If the equation of a surface be transformed from rectilinear to polar co-ordinates, then SffV becomes off, Vr2 cos p. (L. C. D. 531; Lagr. Berl. Mem. an. 1773.) (29.) Conditions which render a curve quadrable. 1 29 Let fy=X, and let X be resolved into X1+X, in which X is an algebraic function of x, and X a function involving transcendants, which however vanishes when x=a, x=b, &c. and let 3 in which ≈=~+X ̧(x − a) (x − b) &c. X, not containing any factors of the first degree, then if the curve is quradrable the limits xa, x=b, &c. (30.) Conditions which render a curve rectifiable. Se (1 + p2) $ =Q, then p= { 1 + (d2 Q) ° } } * px by eliminating a between the equations (1), the ordinate of a rectifiable curve will be obtained from the first, and the expression for the arc of that curve, from the second. TRAJECTORIES. (L. C. D. 735—7.) (31.) Let p(x,y,a)=0 be the equation of a system of curves, of which a is the variable parameter, and let the derived equation be M+Nd,y=0; then by eliminating a between these equations, we obtain (1) (d ̧y,x,y)=0. |