Cor. Hence two infinite series of indefinitely small quantities of the first order, such that each term of the one differs from the corresponding term of the other by a quantity of the second order, are equal. This explains why in Arts. 97, 99, 101 and 102 we have neglected one infinite series and retained another: this is done when the first series is of a higher order than the second. 3. As an illustration of these principles we will give proofs of Guldin's theorems. One theorem is, that the volume, generated by the complete revolution of a plane area about any straight line in its plane and not cutting it, is equal to that of a right cylinder whose section is the plane area and height the length of the path described by the centre of mass of the area. Draw a number of straight lines at right angles to the line AB, about which the revolution takes place, dividing the area S into n strips of equal breadth. Let Pp, Qq be two consecutive lines of this system, typical of the rest, M, N the points where they meet AB. Draw PR, pr perpendicular to Qq. The volume, generated by the revolution of PRrp about AB, differs from that generated by PQqp by the volumes generated by the two curvilinear triangles PQR, pqr. But when n is increased indefinitely, the breadth only of the rectangle is diminished indefinitely, whereas both length and breadth of each triangle is diminished indefinitely; the volumes generated by the latter are therefore of a higher order than that generated by the former. Hence the total volume generated by the area equals the sum of the volumes generated by the rectangles of which Pr is a type, i.e. =Σ (TPM2. MN-πрм3. MN) Also the sum of the areas of the rectangles is the area of the figure S; and since they differ by the sum of the areas of the triangles PQR, pqr, &c., which are of a higher order than the rectangles, the centres of mass of the sum of the rectangles and the figure S must be coincident. Therefore, the distance of the C. M. of S from AB .. volume generated by SS. 2πx. The second theorem is, that the area of the surface, generated by the revolution of a curve about any straight line in its plane and not cutting the curve, is equal to the rectangle, whose length is the length of the curve and breadth the distance of the curve's centre of mass from the straight line. Let PQ be a side of a polygon, either inscribed within or circumscribed about the curve: let R be the middle point of PQ, and therefore its centre of mass. Draw RK perpendicular to the line AB, about which the curve revolves. As in Art. 99, it can be shewn that the area of the surface generated by the revolution of PQ about AB is . 2π PQ. RK. Therefore the total surface generated by the revolution of the polygon about AB =Σ (2πPQ.RK) =2πΣ (PQ. RK) =2πxx perimeter of polygon, where x is the distance of its centre of mass from AB. When the lengths of the sides of the inscribed and circumscribed polygons are diminished indefinitely and their number increased indefinitely, their perimeters differ by indefinitely small quantities, and their centres of mass become coincident. The surfaces generated by each are therefore equal. It is assumed as axiomatic, that as the perimeter of the curve lies in position between the two polygons which ultimately coincide, it is equal to the perimeter of either polygon, its centre of mass coincides with that of either polygon, and the surface generated by it is equal to that generated by either polygon. Hence the surface generated by the curve is equal to the product of the length of the curve into the length of the path traced out by its centre of mass. Each of Guldin's theorems can easily be extended to the case in which the revolution is not a complete one. There is no limitation in either as to the number of times, in which a straight line at right angles to AB cuts the generating curve. Ex. Find the volume and surface of an anchor-ring, the figure generated by the revolution of a circle about a line in its plane, and not intersecting it. Ans. Vol. 22a2c, surface=472ac, where a is the radius of the circle, and c the distance of its centre from the line. 4. The following proposition has been assumed throughout. The limit of |