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 35
Page 8
... cuts Bb in M. Join CM . Then since the points C , M are both in the planes CAd , CPc , CM being produced will meet the curve in c , and cut the axis AQ in R. Now if the circle BPb move parallel to itself , the intersection Pp will also ...
... cuts Bb in M. Join CM . Then since the points C , M are both in the planes CAd , CPc , CM being produced will meet the curve in c , and cut the axis AQ in R. Now if the circle BPb move parallel to itself , the intersection Pp will also ...
Page 9
... cutting plane , be parallel to the axis AQ or B a = 90 ° , a = ∞ , and b , and PM becomes 1 Cc . Hence substituting in the general equation , we have y2 = px . The section is a parabola , similar to , because of equal para- meter with ...
... cutting plane , be parallel to the axis AQ or B a = 90 ° , a = ∞ , and b , and PM becomes 1 Cc . Hence substituting in the general equation , we have y2 = px . The section is a parabola , similar to , because of equal para- meter with ...
Page 10
... cutting plane with the plane Y'A ' X ' , and P be any point in the curve required . Draw PN plane Y'A'X ' , and meeting it in N , and NM BC , Also draw MN ' AX ' , and A'A 1 BC . x and join PM . Then x ' , y ' , ' ; x , y , z , denoting ...
... cutting plane with the plane Y'A ' X ' , and P be any point in the curve required . Draw PN plane Y'A'X ' , and meeting it in N , and NM BC , Also draw MN ' AX ' , and A'A 1 BC . x and join PM . Then x ' , y ' , ' ; x , y , z , denoting ...
Page 12
... cuts the axis = 45 ° . To find the points of inflexion . day = 2.x3 - 6x = 0 dx2 ( 1 + x2 ) 3 ..0 and x2 3 = 0 , -- or x = 0 , and x = ± √3 and the corresponding values of y are y = 0 , and y d ± √3 4 Take , therefore , AM ' , Am ...
... cuts the axis = 45 ° . To find the points of inflexion . day = 2.x3 - 6x = 0 dx2 ( 1 + x2 ) 3 ..0 and x2 3 = 0 , -- or x = 0 , and x = ± √3 and the corresponding values of y are y = 0 , and y d ± √3 4 Take , therefore , AM ' , Am ...
Page 16
... cuts the axis at an angle of 45o . To find at what angle the curve cuts the axis when x = a maxi- mum , or when x1 , we have dy tan . = = ∞otan . 90 ° . .. the angle required = 90 ° Pr Making AM dx 16 18. ...
... cuts the axis at an angle of 45o . To find at what angle the curve cuts the axis when x = a maxi- mum , or when x1 , we have dy tan . = = ∞otan . 90 ° . .. the angle required = 90 ° Pr Making AM dx 16 18. ...
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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...