Examples and Solutions in the Differential CalculusJ. Weale, 1851 - 162 من الصفحات |
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1+p² 1+x² a²+x² a²x² a²y angle asymptote asymptotic circle axes axis of x centre chord co-ordinates common parabola constant contrary flexure coseco cosx cuts the axis d²u d2u d2u d2u dx2 d2u dy d³u diameter differential coefficient distance dx d2u dx du dx dx dx dx dy d2u dx² dx2 dy2 dy dx dy dy e²x² eliminate ellipse equal evolute find the equation frustrum function given Hence hyperbola Inscribe the greatest latus rectum locus logarithmic Maclaurin's Theorem maxima and minima maximum or minimum normal NP=y origin osculating circle perpendicular point of contrary polar equation positive prove radius of curvature right-angles secx semi-cubical parabola sinx spiral straight line substituting subtangent tangent Taylor's theorem trace the curve triangle U₁ variable vertex
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الصفحة 139 - ... of the squares of any two conjugate diameters is equal to the difference of the squares of the axes.
الصفحة 74 - Also v — 5 = 8 = —r ; - = 0 ; м is a minimum. dar dy dxdy Ex. 18. Of all triangular pyramids of a given base and altitude, to find that which has the least surface. Let a, b, с be the sides of the base, h the altitude of the pyramid, 0, <jf>, ^ the inclination of the faces to the base.
الصفحة 47 - Shew that _ 2. Within a given parabola inscribe the greatest parabola, the vertex of the latter being at the bisection of the base of the former. 3. Investigate a differential expression for the radius of curvature, and shew that it is identical with Newton's expression, J limit _ fcrc...
الصفحة 78 - P within a given triangle, from which, if lines be drawn to the angular points, the sum of their squares shall be a minimum.
الصفحة 50 - The centres of two spheres (radii r,r2) are at the extremities of a line 2 a, on which a circle is described. Find a point in the circumference, from which the greatest portion of spherical surface is visible. If x and y be the distances of the point from the centres of the two spheres, / гД / гД и = 2тгг, Ir, - — J + 2тгг2 [ r.¿ -- l , 2ar, whence .r = — . Vr?
الصفحة 45 - ... is also given. Whence the segment of the circle containing the angle BEG is also given. Hence F is a given point, and FE a given line, whence E is in the circumference of the given circle about F whose radius is FE. Wherefore E being in two given circles, it is itself their given intersection.
الصفحة 48 - Calculus," which has been just published, relative to a subject of some interest in the history of science. " 394. Given the latitude of a place, and two circles parallel to the horizon ; to find the declination of a body which, in its apparent diurnal motion, will pass from one of them to the other in the shortest time possible.* " Let Z and P be the zenith and pole, and S and S' the required points on the given parallels, having equal polar distances, PS and P S'.
الصفحة 54 - Blanc be 51O1 yards above the level of the sea, what would be its altitude on a globe of 18 inches in diameter?
الصفحة 55 - VL 58. If any two circles, the centers of which are given, intersect each other, the greatest line which can be drawn through either point of intersection and terminated by the circles, is independent of the diameters of the circles. 59. Two equal circles intersect, the lines joining the points in which any straight line through one of the points of section, which meets the circles with the other point of section, are equal. 60. Draw through one of the points in which any...
الصفحة 54 - A tree, in the form of a frustrum of a cone, is n feet long, and its greater and less diameters are a and b feet respectively ; show that the greatest square beam that can be cut out of it is %-. ¡v feet long.