262. COR. 1. Suppose the resultant K of the forces XYZ, acting at the point (xyx), is in the normal to the sur face at that point, so that X = KVd ̧u, Y = KVd1u, Z = KVdu, the equation (4) then takes the form d ̧u (d ̧yd2x — d ̧xd,y)) + d ̧u (d ̧x d ̧2≈ – d ̧ïd ̧2 x) = 0 ...... (8), or substituting A, B, C for the coefficients of du, du, džu, the equation is Adu Bdu + Cd_u = 0. Now the equation to the osculating plane at the point (xyz) is A (x′ − x) + B (y′ − y) + C (≈′ − ≈) = 0 .............. (9), ...... (10), ૪ x'y's being the current co-ordinates of a point in the plane or normal and when the plane (9) contains the normal (10) we have the condition Adu + Вdu + Cd2u = 0. Hence equation (8) expresses that the osculating plane contains the normal: now this is the property of the shortest line between two points on the surface. 263. PROP. To find the pressure on the surface at any R = X cos a + Y cos ẞ + Z cos y + T{d2 x cos a + d3y.cos ß + d2x cos y}. Now if Ρ be the radius of absolute curvature at the point (xyx), and λ, μu, v the angles it makes with the axes, we have cos λ = pd2x, cos μ = pd3y, cos v = pd ̧3× ; - Rds = Xds cos a + Yds cos ẞ + Zds cos y ་ Let be the angle between the radius of absolute curvature, in the direction of which the resultant of the tensions on the extremities of ds acts, and the normal to the surface, then 264. COR. When there are no forces acting on the string, Y = 0, Z = 0, we have so that X = 0, 265. PROP. To find the form of equilibrium when the string is not attached to a surface. The equations (1), (2), (3), will give the equations of equilibrium, by putting R=0: and eliminating T between (1), (2), (4), and also between (1), (3), (4), we have the two following equations to the form of the string. 266. v (d ̧æd3y – d ̧yd ̧3x) = Yd ̧x – Xd ̧y v (d ̧xdx - dxdx) = Zd,x - Xd, ...(B). COR. In the case of gravity, supposing the axis of vertical and measured upwards, we have This is the differential equation to a straight line in the plane xy, so that the chain hangs in a vertical plane. Take this plane for the plane of xx, and the lowest point as origin. We have, since d. T = gd,, and .. T = g (≈ + c), :. x + C = c log, {≈ + c + √ (≈ + c)2 − c2}, 267. COR. 2. The Catenary also possesses the property of being the curve of total minimum tension, supposing gravity alone to act. have Hence to find the curve having the above property, we by the principles of the Calculus of Variations; or when ≈ + c = cds, and this has been just shewn to be the differential equation to the Catenary. N CHAPTER XI. PROBLEMS. 1. GIVEN the magnitudes of two forces which act on a point, and the angle between the lines in which they act; to find the magnitude of their resultant. Let P, Q be the two forces acting upon the point O (fig 81) in the directions OA, OB. Take OA, OB to represent them, and complete the parallelogram OBCA; the diagonal represents their resultant R. Let a = AOB the given angle. Then from the triangle OAC we have 2. OC2 = OA2 - 20A. AC cos OAC + AC2 = 0A2+20A. OB. cos POQ + OB2; .. R2 = P2 + 2 PQ cos a + Q3. Three forces acting on a point are found to balance each other when their directions make angles 105°, 120°, 135° with each other. Find the relation of the forces to each other. Let F1, F2, F3 be the forces respectively opposite to the angles 1050, 120°, 135°. Then by Art. 28 we have |