Solutions of the Cambridge Problems: From 1800 to 1820, Volume 2Black, Young, and Young, Tavistock-Street, Covent-Garden., 1825 - 653 pages |
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Results 1-5 of 52
Page 6
... axes are and 3 a The branch QAq corresponds to the positive values of x ' . From = 0 to 2a 3 = AB we have y imaginary ; but when x ' is taken 2a 3 negatively > , y again becomes real , and describes the branch Q ' Bq ' . ( 7 ) . Let the ...
... axes are and 3 a The branch QAq corresponds to the positive values of x ' . From = 0 to 2a 3 = AB we have y imaginary ; but when x ' is taken 2a 3 negatively > , y again becomes real , and describes the branch Q ' Bq ' . ( 7 ) . Let the ...
Page 8
... axes are α sin . a and a ; the origin of abscissæ being at the extremity of the axis - minor , because a sin , a is necessarily > a . - 10. To trace the curve whose equation is a2 x2 + ( x — b ) 2 = x2 y ' , we first reduce it to a2 + ...
... axes are α sin . a and a ; the origin of abscissæ being at the extremity of the axis - minor , because a sin , a is necessarily > a . - 10. To trace the curve whose equation is a2 x2 + ( x — b ) 2 = x2 y ' , we first reduce it to a2 + ...
Page 17
... axes are unity . For in the equilateral hyperbola p2 = a2b2 b2 cos.2 - a2 sin.20 1 = when a b = 1 cos . 20 a2 b2 p2 - a2 + b2 2 and p2 ( the perpendicular ) = .. p2 = cos . 24 , ( See equat . a ) . See Prob . 1 , Vol . II . 19 . To find ...
... axes are unity . For in the equilateral hyperbola p2 = a2b2 b2 cos.2 - a2 sin.20 1 = when a b = 1 cos . 20 a2 b2 p2 - a2 + b2 2 and p2 ( the perpendicular ) = .. p2 = cos . 24 , ( See equat . a ) . See Prob . 1 , Vol . II . 19 . To find ...
Page 42
... axes BC , CN'are b and respectively . The point N ' always falls within the given ellipse , and may be found geometrically , by bisecting the ACB by the line CP ' , and drawing the ordinate P'N ' . 39 . The problem may be generalized ...
... axes BC , CN'are b and respectively . The point N ' always falls within the given ellipse , and may be found geometrically , by bisecting the ACB by the line CP ' , and drawing the ordinate P'N ' . 39 . The problem may be generalized ...
Page 47
... axes are = a . Here we may remark , that as most known curves are defined by co - ordinates originating in the vertex , or other points symme- trically placed with respect to the branches , in investigating the nature of a curve , it ...
... axes are = a . Here we may remark , that as most known curves are defined by co - ordinates originating in the vertex , or other points symme- trically placed with respect to the branches , in investigating the nature of a curve , it ...
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Common terms and phrases
abscissa accelerating force altitude angular axis b₁ base bisected body centre of gravity chord circle co-declination co-ordinates cone curve cycloid cylinder denote density descending diameter distance dy dx earth ecliptic ellipse equal equation fluid given point gives Hence horizon hyperbola inclination intersection latitude latus rectum length locus logarithmic spiral moving force orbit ordinate orifice oscillation parabola paraboloid parallel perpendicular plane position problem Prop question radius ratio right angles right ascension shew sides specific gravity sphere spherical straight line substituting subtangent supposing surface tangent triangle velocity vers vertex vertical Vince weight whence whole
Popular passages
Page 654 - ... line and the extremities of the base have the same ratio which the other sides of the triangle have to one...
Page 654 - IF a straight line be divided into two equal, and also into two unequal parts; the squares of the two unequal parts are together double of the square of half the line, and of the square of the line between the points of section.
Page 654 - BAC is cut off from the given circle ABC containing an angle equal to the given angle D : Which was to be done. PROP. XXXV. THEOR. If two straight lines within a circle cut one another, the rectangle contained by the segments of one of them is equal to the rectangle contained by the segments of the other.
Page 654 - In any right-angled triangle, the square which is described upon the side subtending the right angle, is equal to the squares described upon the sides which contain the right angle.
Page 657 - B. less by 1 1 chains than the length of the sewer ; the expense of making it amounted to as many pounds per chain, as there were chains in the street leading to A. The sewer, however, being insufficient to carry off the water, an additional drain was made from a point in this street distant 4 chains from the bridge A., which entered the river at the same point with the sewer, and was equally inclined to the river and sewer. Now it was found that a drain down the middle of each street, at the rate...
Page 693 - Upon comparing the observations with each other, it was discovered that in both the fore-mentioned stars, the apparent difference of declination from the maxima was always nearly proportional to the versed sine of the sun's distance from the equinoctial points. This was an inducement to think that the cause, whatever it was, had some relation to the sun's situation with respect to those points.
Page 713 - This is the same as saying that when a ray of light passes out of one medium into another, the...
Page 685 - W its weight in water, its weight in vacuo will be, 1 — m 6. Three globes of the same diameter and of given specific gravities, are placed in the same straight line. How must they be disposed that they may balance on the same point of the line in vacuo and in water ? 7. If a homogeneous hemisphere, floating in a fluid, be slightly inclined from the position of equilibrium...
Page 658 - A ship, with a crew of 175 men, set sail with a supply of water sufficient to last to the end of the voyage ; but in 30 days the scurvy made its appearance, and carried off three men every day ; and at the same time a storm arose which protracted the voyage three weeks. They were, however, just enabled to arrive in port without any diminution in each man's daily allowance of water. Required the time of the passage, and the number of men alive when the vessel reached the harbor.
Page 655 - A number of persons purchased a field for £345. The youngest contributed a certain sum, the next £5 more, the third £5 more than the second, and so on to the oldest. For the greater accommodation of the seniors, the field was divided into two parts, the younger half taking a portion proportional to the sum they had subscribed ; and in order that each might have an equal share in this portion, they agreed to equalize their contributions, and each to pay ,£22. Required the number of persons and...