## Examples of the Processes of the Differential and Integral Calculus |

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angle arbitrary constant assume asymptote axes becomes Bernoulli bºy branches Chap circle co-ordinates condition cone conjugate point corresponding Crelle's Journal curve cuts the axis cycloid determine differential coefficients differential equation dºw eliminate ellipse equal Euler evolute expression factor find the value formula fraction function geometrical given equation gives Hence hypocycloid independent variable infinite Integrating with respect John Bernoulli Let the equation lines of curvature locus logarithmic logarithmic spiral maximum or minimum Mémoires Multiply negative origin parabola perpendicular plane of reference positive radius roots ſaw ſda Sect singular points singular solution ſº spiral subtangent surface tangent plane theorem triangle vanish whence

### Popular passages

Page 147 - ... 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...

Page 110 - 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

Page 134 - 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...

Page 147 - 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...

Page 130 - 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.

Page 460 - 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.

Page 470 - 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.

Page 130 - To find a point within a triangle from which if lines be drawn to the angular points their sum may be the least possible.

Page 439 - ... 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...

Page 435 - 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...