Examples of the processes of the differential and integral calculus |
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a₁ A₂ a² b2 a²)³ a²x² angle arbitrary constant assume asymptote becomes branches C₁ c²x² circle co-ordinates condition conjugate point curve cycloid determine differential coefficients differential equation dx dx dx dy dx dx² dy dx dy dy dy eliminating ellipse equal Euler factor formula function given equation gives Hence hypocycloid infinite Integrating with respect intersection John Bernoulli Let the equation lines of curvature locus logarithmic logarithmic spiral maximum Mémoires minimum Multiply negative origin parabola perpendicular plane of reference radius singular points singular solution spiral subtangent surface tangent plane theorem vanish whence x²)³ y²)³
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Page 141 - Cum autem ob proprietatem tam singularem tamque admirabilem mire mihi placeat Spira haec mirabilis, sic ut ejus contemplatione satiari vix queam; cogitavi, illam ad varias res symbolice repraesentandas non inconcinne adhiberi posse. Quoniam enim semper sibi similem...
Page 104 - 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 i - A Treatise on the Application of Analysis to Solid Geometry. Commenced by DF GREGORY, MA, late Fellow and Assistant Tutor of Trinity College, Cambridge ; Concluded by W. WALTON, MA, Trinity College, Cambridge.
Page 128 - Examples, p. 130. 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 curve may be defined as the locus of a point in a line which slides on and revolves about a fixed point, while the distance between the generating point and a fixed right line on either side of it is constant. The curve is constructed as follows: Let 0 be the fixed point, XX...
Page 141 - ... sobolis parentibus per omnia similis Emblema; Simillima Filia Matri: vel (si rem aeternae veritatis Fidei mysteriis accomodare non est prohibitum) ipsius aeternae generationis Filii, qui patris yelut Imago, et ab illo ut Lumen a Lumine emanans eidem 6{i6<3ios existit, qualiscumque adumbratio.
Page 124 - TRIANGLE, FROM WHICH IF LINES BE DRAWN TO THE ANGULAR POINTS, THE SUM OF THEIR SQUARES SHALL BE A MINIMUM.
Page 141 - Quoniam enim semper sibi similem et eandem spiram gignit, ntcumque volvatur, evolvatur, radiet; hinc poterit esse vel sobolis parentibus per omnia similis Emblema; Simillima Filia Matri: vel (si rem aeternae veritatis Fidei mysteriis accomodare non est prohibitum) ipsius aeternae generationis Filii, qui...
Page 141 - Aut, si mavis, quia curva nostra mirabilis in ipsa mutatione semper sibi constantissime manet similis et numéro eadem, poterit esse vel fortitudinis et constantiae in adversitatibus ; vel etiam carnis nostrae, post varias alterationes et tandem ipsam quoque mortem, ejusdem numéro resurrecturœ symbolum; adeo quidem ut si Archimedem imitandi hodienum consuetude obtineret libenter spiram hanc túmulo meo juberem incidí cum épigraphe : Eadem mulata resurget.
Page 425 - Ex. 2. The axes of two equal right circular cylinders intersect at right angles. Required the volume common to the cylinders. Let OA and OB (fig.
Page 124 - To find a point within a triangle from which if lines be drawn to the angular points their sum may be the least possible.