COR. 8. If, however, we have the two equations Ap=f. (0) or R2=0, and Ap=ƒ3. (0) or R=0, then the spiral will admit a tangent. For, in that case, we shall have Now, if R, be positive, and (1) sufficiently diminished, both these series will be positive; and, therefore, SR and SR will be greater than SQ and SQ respectively; and, therefore, RPR' will be a tangent. COR. 9. In a similar way it. may be shown, that, if the number of the coefficients R2, R3, R, &c., which vanish at any point, be odd, the curve does not admit a tangent, and that, if the number be even, it does. COR. 10. Hence, if two spirals, referred to the same pole, have at any point a common tangent, the first differential coefficients of their radii vectores must, at that point, be equal. For at that point, since their radii vectores, and their tangents coincide, (p) dp and and, consequently, de p.do in both. d Ꮎ must be the same COR. 11. Hence, if two spirals, referred to the same pole, touch each other internally, dp is the same d Ꮎ in both. For in this case the spirals have a common tangent. DEF. A spiral is said to be concave at any point when the parts of it, immediately adjacent to that point, lie between the pole, and the tangent to that point. DEF. A spiral is said to be convex at any point, when the tangent to that point lies between the pole and the arcs, immediately adjacent to that point. DEF. If, while the spiral angle increases, the spiral from being convex becomes concave, or from being concave becomes convex, the point, where the change takes place, is called a point of contrary flexure. If, at any point, the radius vector of a spiral becomes its tangent, it is manifest, that, at this point, a change from convexity to concavity takes place; but, since the spiral angle does not increase during the change, this point is not a point of contrary flexure. PROP. LI. To find when a spiral is convex, and when concave. Using the same figure, construction, and notation as before, it appears, that Now if the curve be concave, Q lies between R and S, and SR is greater than SQ; and, therefore, the series (D) is positive; hence, R, the coefficient of the first term, must be positive; (unless it=0). But R2=A2p-ƒ2. † (0) ; d2p and therefore, ƒ2¢(0) or is less than Ap, In the same manner it may be shown, that, when the de COR. 1. Conversely. cave, or convex according as is less or greater Ex. 1. To find when the logarithmic spiral is concave, and when convex. Here the equation is pao; Now, a must always be less than (24+ 1 ) ao ; and, therefore, the curve will always be concave, and admit no point of contrary flexure. Ex. 2. To find when the parabola is concave 2 dp dp2 pdo do 2 pdo2 + p = 2 a. t° (1+t3) +a. (1+t2) = a (1 + 3 t2 + 2t4). and, therefore, the curve is always concave. Ex. 3. To find when the Lituus, (a spiral whose 4 1 2 and ... is less than ; hence, 02 is greater than; and, therefore, (8) is greater than 1 conversely, if (6) be greater than the spiral is concave; in the same manner it may be shown, that 1 if (0) be less than , the spiral is convex. Hence it 2 appears, that there is a point of contrary flexure, |