Mathematical Questions and Solutions, Τόμος 42F. Hodgson, 1885 |
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Σελίδα vii
... tangent planes be drawn , then the centre of mean position of their points of contact is fixed .... 6456. ( G. Heppel , M.A. ) If the expansion of sec V2 x + & c . , and that of 2 sec2 x be 2+ 2 ! 1 + И 22+ 4 ! 2 ! 1 50 be x2 + x4 + & c ...
... tangent planes be drawn , then the centre of mean position of their points of contact is fixed .... 6456. ( G. Heppel , M.A. ) If the expansion of sec V2 x + & c . , and that of 2 sec2 x be 2+ 2 ! 1 + И 22+ 4 ! 2 ! 1 50 be x2 + x4 + & c ...
Σελίδα viii
... tangents to a circle and DE bisects them ; show ( 1 ) that DE cannot meet the circle ; and hence prove ( 2 ) that for acute angles - sin tan 0-0 . ་ 99 7170. ( The Editor . ) - If 10 cards are taken at random from a pack , show that the ...
... tangents to a circle and DE bisects them ; show ( 1 ) that DE cannot meet the circle ; and hence prove ( 2 ) that for acute angles - sin tan 0-0 . ་ 99 7170. ( The Editor . ) - If 10 cards are taken at random from a pack , show that the ...
Σελίδα ix
... ellipse = 3 ( 1 + 2e - e2 ) 2 : 8 . - 29 7385. ( Professor Wolstenholme , M.A. , Sc.D. ) In an equilateral triangle ABC is inscribed a circle , any tangent to this circle meets the sides CB , CA in the points A ' , CONTENTS . ix.
... ellipse = 3 ( 1 + 2e - e2 ) 2 : 8 . - 29 7385. ( Professor Wolstenholme , M.A. , Sc.D. ) In an equilateral triangle ABC is inscribed a circle , any tangent to this circle meets the sides CB , CA in the points A ' , CONTENTS . ix.
Σελίδα x
... tangent , show that the locus of its intersection with the tangent will be a circle which touches or falls entirely without the ellipse according as cos a is less or greater than the excentricity of the ellipse ......... 72 7514 ...
... tangent , show that the locus of its intersection with the tangent will be a circle which touches or falls entirely without the ellipse according as cos a is less or greater than the excentricity of the ellipse ......... 72 7514 ...
Σελίδα xi
... tangents at P , Q , R form a triangle P'Q'R ' ; prove that the ratio k : 1 of the triangles PQR , P'Q'R ' is given by 2 - { k2 + ( k − 4 ) a2X2 + b ̈Y2 } * = ( ≥4 — k ) ( a2X2 – b2X2 ) 3 .. - 7532. ( Â . Mukhopâdhyây . ) — Prove that ...
... tangents at P , Q , R form a triangle P'Q'R ' ; prove that the ratio k : 1 of the triangles PQR , P'Q'R ' is given by 2 - { k2 + ( k − 4 ) a2X2 + b ̈Y2 } * = ( ≥4 — k ) ( a2X2 – b2X2 ) 3 .. - 7532. ( Â . Mukhopâdhyây . ) — Prove that ...
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Συχνά εμφανιζόμενοι όροι και φράσεις
a₁ a²X² algebraic Axiom BIDDLE bisects Brocard Brocard circle Brocard point C₁ cards centre of similitude centroid chance chord circle circumcentre circumcircle coefficient coincide conic coordinates cos² cosec curve diagonals distance drawn ellipse equal equation fixed point focus given H₂ HANUMANTA RAU hence horizontal hyperbola inscribed intersection inverse limaçon line joining locus mid-point nine-point circle opposite P₁ parabola parallel perpendicular plane points of contact polar position probability prove R₁ radical axis radii radius respectively right angles roots sides Similarly sin² Solution by W. J. C. squares straight line symmetrical T. C. SIMMONS tangent triangle ABC trilinear coordinates values velocity vertex vertical W. J. C. SHARP whence
Δημοφιλή αποσπάσματα
Σελίδα 131 - When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it.
Σελίδα 131 - A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line.
Σελίδα 137 - Now the question is, whether, if this story were related to the wild boy caught some years ago in the woods of Hanover, or to a savage without experience, and without instruction, cut off in his infancy from all intercourse with his species, and, consequently, under no possible influence of example, authority, education, sympathy, or habit; whether, I say, such a one would feel, upon the relation, any degree of that sentiment of disapprobation of Toranius's conduct which we feel, or not?
Σελίδα 137 - Toranius's conduct which we feel, or not. They who maintain the existence of a moral sense ; of innate maxims ; of a natural conscience ; that the love of virtue and hatred of vice are instinctive ; or the perception of right and wrong intuitive, (all which are only different ways of expressing the same opinion,) affirm that he would. They who deny the existence of a moral sense, &c. affirm that he would not. — And, upon this, issue is joined.
Σελίδα 132 - If a straight line meets two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles...
Σελίδα 136 - ... every one who possesses prudence ;" and you will have the celebrated argument of Aristotle, Eth. sixth book, to prove that the virtues are inseparable ; viz. He who possesses prudence, possesses all virtue ; He who possesses one virtue, must possess prudence; therefore He who possesses one, possesses all.
Σελίδα 137 - Having experienced, in some instance, a particular conduct to be beneficial to ourselves, or observed that it would be so, a sentiment of approbation rises up in our minds ; which sentiment afterwards accompanies the idea or mention of the same conduct, although the private advantage which first excited it no longer exist.
Σελίδα 136 - X : eg Prudence has for its object the benefit of individuals ; but prudence is a virtue; therefore, some virtue has for its object the benefit of the individual, is part of Adam Smith's reasoning (Moral Sentiments) against Hutcheson and others, who placed all virtue in benevolence.
Σελίδα 130 - Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another.
Σελίδα 22 - A thin uniform spherical cap being supposed to attract according to the law of the inverse fifth power of the distance...