–ydy - po con (a? – y2) Jo ait, (a? - y2)} = x; x2 + y2 = a. Ex. 3. If one = y, and x = 0, and any = 0, when y=a, then y = {este å}. 176.] Geometrical problems depending on the integration of differential quantities. Ex. 1. Find the curve of which the subnormal is constant. Taking the value of the subnormal given in (41), Art. 219, Vol. I, we have dx and taking the origin to be on the curve, we have y2 = 2ax. Ex. 2. Find the curve, of which the subtangent is constant. y = ea. Ex. 3. Find the equation to the curve, of which the tangent is constant. Taking the value of the tangent given in (42), Art. 219, Vol. I, we have y*(1+ dietea; if we suppose y to decrease, as x increases. Let us also assume y = a, when x = 0; then which is the equation to the equitangential curve; see Vol. I, Art. 200, equation (27). Ex. 4. Find the equation to the curve whose normal is constant. In this case, by (42), Art. 219, Vol. I, y? (1 + .) = a?; . dx = $ydy; (a-y) and assuming x = 0, when y = a, we have xo + y2 = a?. Ex. 5. Find the equation to the curve all the normals to which pass through the same point (a, b). This property is expressed by the equation (x — a) dx + (y—b) dy = 0; i (x — a)+(y—b)2 = c; which is the equation to a circle, whose radius is c. Ex. 6. Find the curves in which the sum of (1) the abscissa and the subnormal, (2) the ordinate and the subnormal, is constant. (1) (x— a)2 + y2 = c2; (2) x+y+ a log (a-y) = c. Ex. 7. Find the curve in which the tangent is equal to the radius vector of the point of contact. so that the required line is either straight, or an equilateral byperbola, according as the lower or the upper sign is taken. Ex. 8. Find the curve which cuts all its radii vectores at the same angle. do Let this angle be tan-1c; then raho = c; Ex. 9. Find the curve in which the perpendicular on the tangent is equal to the part of the tangent intercepted between the foot of the perpendicular and the point of contact. This is a particular case of the preceding example ; that, viz., in which c=1; so that the equation of the curve is r = aeo. It may also be solved in the following way; by the condition we have pa = -p; so that y2 = 2pa; 1 u“ + dozi ... r = aee. Ex. 10. The curve in which the polar subtangent is constant is the reciprocal spiral. Ex. 11. The curve in which the polar subnormal is constant is the spiral of Archimedes. Ex. 12. Find the curve in which the perpendicular from the origin on the tangent is equal to the abscissa. p = r cos 0 ; du? .. p = 2 a cos 0. Ex. 13. Find the curve of which the radius of curvature is equal to a. r dr Ex. 14. If p = ar, the equation to the curve is 2 a Ex. 15. If the angle between the radius vector and the curve = 0, the curve is a circle. Ex. 16. If the subnormal is equal to the abscissa, the curve is a hyperbola. Ex. 17. If the radius of curvature = p, the curve is the involute of the circle. CHAPTER VII. THE APPLICATION OF SINGLE DEFINITE INTEGRATION TO THE THEORY OF SERIES. SECTION 1.- On the Convergence and Divergence of Series. 177.] The subject of series has come under our notice many times in the preceding parts of our treatise, and in various forms. At one time processes have been investigated for the development of functions into series; as Maclaurin's and Taylor's theorems, and others subordinate to and cognate to them; at other times, as in the Integral Calculus, the sums of series have been determined : that is, functions have been found of which the given series are the developments. These two processes are evidently inverse to each other. Now, when a function is developed into a series, that series taken to a finite number of terms, or even to an infinite number of terms, cannot always be assumed to be, and used as, the equivalent of the function; it is necessary to calculate what has been called “the remainder" in the theorems of Maclaurin and Taylor: and it is only when this remainder fulfils certain conditions, that the series can be used as an adequate equivalent of the function. Similarly there are certain conditions which a series must satisfy when it is adequately represented by a given function, or when in other words, the series can be summed, and that function is the sum; and there are also certain limits of the variable within which the summation is possible, and outside of which it is impossible. These thus far have not been discussed; and I have assumed only that knowledge of the subject which will be found in ordinary treatises on algebra. The subject however requires more complete and precise investigation, and arises now in the regular course of our Treatise on the Integral Calculus; because single definite integration supplies a theorem on convergence and divergence of series, which determines whether a given series is capable of summation or not; and which is of wider application than any heretofore suggested. It is indeed one of the most important applications of the calculus |