...represent the rectangular co-ordinates of any point in the tangent to the point Y' x ; and T the tangent of the angle which the tangent to the curve makes with the axis of x, Then we shall have, Gregory's examples on the differential and integral calculus, page 142. We have...

...the instance in the text, this often affords a ready method of elimination. 198. If we denote by T the angle which the tangent to the curve makes with the axis of x, we have _d*ydx-d*xdy • . Q,T - - J - g - J - " - J dy" + dx* <fr _ dyd*x - dxd*y •'' fa~- ~~3?...

...£S= a (seco — taño), тг and this value approximates to 0 as в approaches to — . 4. Let ty be the angle which the tangent to the curve makes with the radius vector, then tani/r = — =cos0 .................... (4). When the radius vector coincides with...

...a fixed point, p the radius of curvature ; then the last equation gives 4*+a*!-* 1 as 2 as r Let $ be the angle which the tangent to the curve makes with the axis of x ; and assume Oj = & sin /3, G 2 = k cos /3 ; thus Jc p = ^ cos (£-£). This is the well-known intrinsic equation...

...X,\i X~J ' respectively, where в is a cube root of unity ; and since dxt dx — idy where •$• is the angle which the tangent to the curve makes with the axis of x, we hence infer that the three curves of the system which pass through a point cut each other at angles...

...y — if' = -- (x — x') (3.9), the coefficient -- being the negative reciprocal of the tangent of the angle which the tangent to the curve makes with the axis of x. 281. Con. — The tangent of the, angle, which a normal to a curve at a particular point makes with...

...represents the tangent to an hyperbola referred to rectangular asymptotes, — —, is the tangent of the X angle which the tangent to the curve makes with the axis of x ; and 2y' is the distance from the origin at which the tangent cuts the axis of y, as in all equations of...

...assumes the values dxt respectively r where 6 is a cube root of unity ; and since dxt dx - idy where ^ is the angle which the tangent to the. curve makes with the axis of a;, we hence infer that the three curves of the system which pass through a point cut each other at...

...the point R from this line is a(l — sin 6), since RS = a(scc 6 — tan 6), and this value Let \|r be the angle which the tangent to the curve makes with the radius vector (or vector, as we may more briefly call it), then tan ->/r = —— = cos 6 (a) When...

...the absolute temperature of the triple point (<tp\ =hi ' But dpfdr is the trigonometrical tangent of the angle which the tangent to the curve makes with the axis 00, and as the latent heats and differences of specific volume are in general different for the three...