...partly concave towards the axis of #, and partly convex. The distance from the axis of the highest We are here concerned with the concave portion. If...curvature at any point of the concave portion we have V(l+/) pa Thus at the points P and Q since the curve touches the straight lines we have 3a p ~ a '...

...+ (n * ~ 1} { log ^T — 4/(M) — l } KC ° S Jt or, neglecting 4>f(n), + *^- an cos n + fl. sin m If p be the radius of curvature at any point of the central line of vortex core, we can easily prove that DO/ (L -7 = - + — -s- K cos n-^r + &, sin nty),...

...traced on a piece of paper, which is then folded so as to become a cylinder of radius b; shew that, if , p be the radius of curvature at any point of the curve which the circle now becomes, then -*—-, + r, cos4 - , where s is the distance, paoa measured...

...lengths of the semiaxea. 3. Determine the equation of the inverse of the curve In + mr.i + nrs = O, 4. If p be the radius of curvature at any point of the lemniscate n rs = a 2 , prove that p (ri — r^) is constant. j. Determine the envelope of an arbitrary...

...traced on a piece of paper, which is then folded so as to become a cylinder of radius b; shew that, if p be the radius of curvature at any point of the curve which the circle now becomes, then — 5 = — 5 + r5 cos4 -, where s is the distance, measured...

...another on the inside ; find the hodograph of a point on the circumference of the rolling circle. 16. If p be the radius of curvature at any point of the hodograph of a central orbit, and p the perpendicular from the pole of the hodograph on the tangent...