...the straight line joining this and the proposed point of the curve will be the tangent. 339. Con. 2. The length of the tangent intercepted between the point of contact and the plane of %y is equal to when o?' = 0, and is therefore expressed by V^ + o^+ar — = л л/! 4 (dJLV...

..., _ a dsJ a ds = V(dx* + dy). . (0 (2) FIG. 33. 165. The distance PT (Fig. 34) on the tangent line intercepted between the point .of contact and the axis of x is sometimes called the tangent, and in like manner the intercept PN is called the normal. From the triangles...

...as л and ds = \f(dS + dy1). . . (2) ° FIG зз 165. The distance PT (Fig. 34) on the tangent line intercepted between the point of contact and the axis of x is sometimes called the tangent, and in like manner the intercept PN is called the normal. From the triangles...

...tractriz or equitangential curve. If AB is a curve such that PT, the length of the intercepted tangent between the point of contact and the axis of x, is...then the locus of P is the equitangential curve. Let P and Q be two consecutive points on the curve; let (x, y) be the point P, and OA = PT = a. Then a...

...equation of the tangent. 136. The Subtangent. Def. The projection on the axis of X of that portion of the tangent intercepted between the point of contact and the axis of X is called the subtangent. To find where the tangent intersects the axis of X, we make y = 0 in the equation...

...foot of the ordinate on the tangent is of constant length. 11. In the tractory prove that the portion of the tangent intercepted between the point of contact and the axis of X is of constant length. 201. Polar Co-ordinates. If the equation of the curve be referred to polar co-ordinates,...

...foot of the ordinate on the tangent is of constant length. 11. In the tractory prove that the portion of the tangent intercepted between the point of contact and the axis of x is of constant length. 91. Polar Co-ordinates. If the equation of the curve be referred to polar coordinates,...

...90° - ¿NPT = Z.NTP = Ц> dy 1 dx tan \1> = j .'. cot \1> = — = — dx dy_ ay dx (i) The portion of tangent intercepted between the point of contact and the axis of x is called the lengtL of the tangent. л In the rt. La APTN, Length of tangent = PT = PN cosec у = PN...

...(o, a). Prove that the equation of the tangent at « is X + Y sinh u = ait, and hence shew tha- tho length of the tangent intercepted between the point of contact and the axis of x is constant. 6. Prove the formula for integration by parts. . 9 Integrate С M 7. Integrate ез> Integrate...

...cos y \ y=i(*> y\~x\ dx)x~*\ y=y\ 1 + -,2 dx ) У=У\ J 2. Length of tangent: N SX *• X The portion of the tangent intercepted between the point of contact and the axis of x is called the length of the tangent. In A PTN . Length of tangent = PT = PN • cosec у -\¡] + = PN...