Examples of the Processes of the Differential and Integral CalculusJ. and J.J. Deighton, 1846 - 529 стор. |
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Сторінка iii
... respect it will be seen to agree with Professor Peacock's Collection of Examples ; and indeed if a second edition of that excellent work had been published I should not have undertaken the task of making this compilation . But as ...
... respect it will be seen to agree with Professor Peacock's Collection of Examples ; and indeed if a second edition of that excellent work had been published I should not have undertaken the task of making this compilation . But as ...
Сторінка 28
... respect to a are transformed into those of a with respect to y by means of the formulæ , dy 1 dx dx = dy d2 x d'y dy2 > d.c2 da dy 2 dx dx ( d ) - dy dy 3 dy dx3 \ dy2 dx 5 " dy and similarly for higher orders . The reader will find the ...
... respect to a are transformed into those of a with respect to y by means of the formulæ , dy 1 dx dx = dy d2 x d'y dy2 > d.c2 da dy 2 dx dx ( d ) - dy dy 3 dy dx3 \ dy2 dx 5 " dy and similarly for higher orders . The reader will find the ...
Сторінка 45
... respect to a , diz d'y dx2 = b dx2 Eliminating b , we have ď z and dx3 = dy b dx3 ď3 z ď3y dz d3y - = 0 . dx3 dx2 dx2 dx3 This is the condition that a curve in three dimensions should be a plane curve . ( 11 ) Eliminate the exponentials ...
... respect to a , diz d'y dx2 = b dx2 Eliminating b , we have ď z and dx3 = dy b dx3 ď3 z ď3y dz d3y - = 0 . dx3 dx2 dx2 dx3 This is the condition that a curve in three dimensions should be a plane curve . ( 11 ) Eliminate the exponentials ...
Сторінка 46
... respect to a only , dz dz = y ( y ) ; and therefore a ≈ = 0 . dx dx ( 16 ) Eliminate the function from the equation - - y Nz = $ ( x − mx ) . Differentiating with respect to a only , dz - - n dx = p ' ( x − mx ) ( 1 dz - -m dx ...
... respect to a only , dz dz = y ( y ) ; and therefore a ≈ = 0 . dx dx ( 16 ) Eliminate the function from the equation - - y Nz = $ ( x − mx ) . Differentiating with respect to a only , dz - - n dx = p ' ( x − mx ) ( 1 dz - -m dx ...
Сторінка 47
... respect to x , dz dx ( 1 ) = nx ” −1 ¢ - Yxn- -2 a yn + +1 ( 00 a ( 2 ) . Differentiating with respect to y , dz dy ( 2 ) = x2 - 1 φ ' ( 1 ) + + ny " - 1 ↓ X + yr Multiply ( 1 ) by x , ( 2 ) by y and add , d.z dx dz + y = nx . dy This ...
... respect to x , dz dx ( 1 ) = nx ” −1 ¢ - Yxn- -2 a yn + +1 ( 00 a ( 2 ) . Differentiating with respect to y , dz dy ( 2 ) = x2 - 1 φ ' ( 1 ) + + ny " - 1 ↓ X + yr Multiply ( 1 ) by x , ( 2 ) by y and add , d.z dx dz + y = nx . dy This ...
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Загальні терміни та фрази
a₁ a² b2 a²x² angle arbitrary constant assume asymptote becomes branches C₁ c²x² circle co-ordinates condition Crelle's Journal curvature curve cycloid determine differential coefficients differential equation dx dx dx dy dx² dy dy dy dz dy² dz dz eliminate ellipse equal Euler factor formula fraction function gives Hence hypocycloid infinite Integrating with respect intersection John Bernoulli Let the equation lines of curvature locus logarithmic logarithmic spiral Mémoires multiplying negative origin parabola perpendicular radius SECT singular points singular solution spiral Substituting subtangent surface tangent plane theorem triangle vanish whence x²)³
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Сторінка 133 - ... non inconcinne adhiberi posse. Quoniam enim semper sibi similem et eandem spiram gignit, utcunque volvatur, evolvatur, radiet ; hinc poterit esse vel sobolis parentibus per omnia similis emblema...
Сторінка 98 - This is the celebrated problem of the form of the cells of bees. Maraldi was the first who measured the angles of the faces of the terminating solid angle, and he found them to be 109° 28' and 70° 32
Сторінка 120 - AS' is traced a second time ; thus, the curve is traced twice by one revolution of the radius-vector. THE CONCHOID OF NICOMEDES.* 150. This curve was invented by Nicomedes, who lived about the second century of our era, and was, like the preceding, first formed for the purpose of solving the problem of finding two mean proportionals, or the duplication of the cube ; but it is more readily applicable to another problem not less celebrated among the ancients, that of the trisection of an angle. The...
Сторінка 133 - Lumine emanans eidem o.adötoi,- existit, qualiscumque adumbratio. Aut, si mavis, quia Curva nostra mirabilis in ipsa mutatione semper sibi constantissime manet similis et numero eadem, poterit esse vel fortitudinis et...
Сторінка 117 - Find that point within a triangle, from which if lines be drawn to the angular points, the sum of their squares shall be a minimum.
Сторінка 430 - II. (if + z* — x*} p — 12. Required the equation of the surface which cuts at right angles all the spheres which pass through the origin of coordinates and have their centres in the axis of x. It will be found that this leads to the partial differential equation of the last problem.
Сторінка 440 - Find the surface in which the coordinates of the point where the normal meets the plane of xy are proportional to the corresponding coordinates of the surface.
Сторінка 117 - To find a point within a triangle from which if lines be drawn to the angular points their sum may be the least possible.
Сторінка 409 - ... be lengthened. Newton found that the length of the polar must be to that of the equatorial canal as 229 to 230, or that the earth's polar radius must be seventeen miles less than its equatorial radius : that is, that the figure of the earth is an oblate spheroid, formed by the revolution of an ellipse round its lesser axis. Hence it follows, that the intensity of gravity at any point of the earth's surface is in the inverse ratio of the distance of that point from the centre, and consequently...
Сторінка 405 - OY their common perpendicular at their point of intersection 0, and a the radius of the base of each cylinder. Then the figure represents one eighth of the required volume V. A plane passed perpendicular...