19 (cos x1,x)2 + (cos x1,y)2 + (cos x1,≈)2 = 1, (cos x1,x)2 + (cos ≈1,y)2 + (cos ≈1,≈)2 = 1. If both systems are rectangular, then cos x1,x. cos y1,x + cos x1y. cos y1y + cos x1,. cos y1,x=0, cos x1,x . cos ≈1,x + cos x1y. cos z1y + cos x1,. cos ≈1,≈=0, cos y1,x. cos ≈1,x + cos y1,y. cos ≈1,y + cos y1,≈ . cos ≈1,≈ = 0. 19. (31.) Let t represent the trace of the plane ay on a11, then if both systems are rectangular, the position of the new axes may be determined from the three angles cos x=x1 (cos $.cos y=x, (sin .cos = .sin + sin.cos.cos x) -≈, sin þ. sin x, - cosp.sin .cos x) − y1 (sin p.sin + cos p. cos.cos x) + cos . sin a, sin.sinx+y, cosy.sin x + cos x. 1 X If t be supposed to coincide with a1, these formulæ become sin .cos X, y=x1 sin -y, cos.cosx+x, cos p. sin x, x=y, sin x + cos X. +81 If the origin alone be changed, then x = x1+a, y=y1+b, 1 (2) If the position of the origin, as well as the direction of the axes be changed, then a, b, c, the co-ordinates of the new origin must be respectively added to the above values of x, y, and ≈. (32.) The equation to any surface being given, to find the equation to the curve formed by its intersection with a plane; let the equation to the surface be f(x,y,x)=0, and the plane of a1y, the intersecting plane, then the trace of 1 x11 on xy being assumed the axis of x, and a line perpendicular to a1 in the plane a1y, the axis of y1, (2) become 1 =y1 sin X· These values of x, y, z being substituted in the equation to the surface will give the equation to the curve required. (33.) Transformation of rectangular to polar co-ordinates. Let u be the radius vector, u,, its projection on the plane xy, and a, b, c the co-ordinates of the pole, then ≈=u sinu ̧„‚u+c. The radius vector must always be considered positive. THE SPHERE. (34.) Let a, ß, y, be the co-ordinates of the centre; and r the radius; the most general form of the equation is 2{(x-a)(y-ẞ)cos x,y+(x−a)(≈—y)cos x,x+(y−ẞ)(≈—y)cosy,x} 1 +(x − a)2 + (y — ẞ)2 + (≈ − y)2 = r2. (1) If the origin is at the centre, the equation becomes x2 + y2+x2+2(xy cos x,y + xx cos x,x+yz cos y,z) = r2. If the axes are rectangular, (1) becomes If the origin is on the surface of the sphere, then 2 x2+ y2+x2-2 (ax+By+y≈)=0. If the origin is at the centre, then x2 + y2 + x2= p2. The equation to a plane touching the sphere at the point (x1,y11) is (x − a) (∞, − a) + (y − ẞ) (y1 − ẞ) + (x − y) (%1 − y) = r2. If the origin is at the centre, this equation becomes THE CYLINDER. (35.) The general equation to a cylindrical surface, the directrix of which is in the plane xy, is f(x − az, y-b≈) = 0. The equation to an oblique cylinder on a circular base, in the plane xy, the origin being in the circumference of the base, (x − a x)2 + (y — bz)2 = 2r (x — az). is Every section of this cylinder made by a plane inclined to its axis is an ellipse, or a circle. (H.A. G. 331-4.) THE CONE. (36.) Let a, ẞ, y, be the co-ordinates of the vertex, and let the directrix be in the plane xy, the general equation to a conical surface is xy is The equation to a right cone on a circular base in the plane If the origin is at the centre of the base, the equation The equation to a right cone, of which the base is the ellipse The equation to an oblique cone on a circular base, the origin being in the circumference of the base, is (az −yx)2 + (ßx — yy)2 = 2r.(x − y)(ax—yx). If the origin is at the centre of the base, the equation becomes (a ≈ − y x)2 + (ẞx —yy)2 = r2 (x − y)o. (37.) The equation to the intersection of the cone with a plane passing through the origin, perpendicular to the plane xx, and inclined to the plane ry at an angle X, is 2 {(y cos x)2 — (r sin x)2 } y12 + y2x2 +2yr2 sin x·y1 — p2y2=0. γ If tan x<, the plane is inclined to the side of the cone, X and cuts only one sheet of the surface; in this case, the curve is an ellipse. If tan x=1, the plane is parallel to the side of the cone, જ X and the curve is a parabola. γ If tan X> the plane cuts both sheets of the surface, and r the curve is an hyperbola. The equation to the intersection of the oblique cone y3 (x2 + y2)+(a2 — r2) x2 — 2ɑyxx+2ar2x+y2r2 = 0, and a plane situated as above is {(cosx-asinx)-(r sinx)} y,+y2x2+2ar2 cos x· Y1—y2r2=0. The curve is an ellipse, a parabola, or an hyperbola, ac 2 (H. A. G. 335-46; G. G. A. Ch. xv; Biot, Ch. vi.) SURFACES OF THE SECOND ORDER. (38.) The general form of the equation to a surface of the second order is Ax2+By2+Cx2+2(41y≈+B1x≈+С1xy+Ã ̧x+B2y+C2≈)+D=0. If the axes are parallel to any system of conjugate diametral planes, the equation becomes Ax2 + By2 + C≈2 + 2 (A ̧x + B2y + C2≈) +D=0. 2 2 (1) A straight line cannot intersect a surface of the second order in more than two points. A A The equation to a diametral plane, which is the locus of the points of bisection of all chords parallel to the line x=mx, y=nz, is (Am + C1n + B ̧) x + (C ̧m + Bn + A1) y + (B ̧m + A1n + C) ≈ + A ̧m + B2n + C2 = 0. The number of systems of conjugate diametral planes is unlimited; of these, however, only one system can be rectangular. The co-ordinates of the centre are A-BC < 0, B-AC < 0, C2-AB<0, 2 AA12 + BB12 + CC122 — ABC — 2А1B1C1<0. If the surface has not a centre, then (39.) Surfaces of revolution. If the axis of revolution coincides with one of the co-ordinate axes, as x, the general equation to a surface of revolution is y2 + x2=f(x). The conditions which determine a surface of the second order to be a surface of revolution are also A1, B1, C1 must all have the same sign. The equations to the axis of revolution are |