Page:Somerville Mechanism of the heavens.djvu/89

Chap I.] 42. Thus, when a particle of matter urged by any forces whatever remains in equilibrio, the sum of the products of each force by the element of its direction is zero. As the equation is true, whatever be the values of &delta;x, &delta;y, &delta;z, it is equivalent to the three partial equations in the direction of the axes of the co-ordinates, that is to

for it is evident that if the resulting force be zero, its component forces must also be zero.

43. A pressure is a force opposed by another force, so that no motion takes place.

44. Equal and proportionate pressures are such as are produced by forces which would generate equal and proportionate motions in equal times.

45. Two contrary pressures will balance each other, when the motions which the forces would separately produce in contrary directions are equal; and one pressure will counterbalance two others, when it would produce a motion equal and contrary to the resultant of the motions which would be produced by the other forces.

46. It results from the comparison of motions, that if a body remain at rest, by means of three pressures, they must have the same ratio to one another, as the sides of a triangle parallel to the directions.

47. The normal to a curve, or surface in any point m, fig. 12, is the straight line mN perpendicular to the tangent mT. If mm' be a plane curve

x and y being the co-ordinates of m, a and b those of N. If the point m be on a surface, or curve of double curvature, in which no two of its elements are in the same plane, then,

x, y, z being the co-ordinates of m, and a, b, c those of N. The