An Elementary Treatise On Analytical Geometry
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This is the curve which a nail in the rim of a carriage-wheel describes in the air during the motion of the carriage on a level road. The curve derives its name from two Greek words signifying "circle-formed. " The line AL over which the generating circle passes in a single revolution is called the base of the cycloid, and if I be the middle point of AL, the point K is called the vertex, and the line KI the altitude or axis of the curve. To find its equation, Fig. 14.
CL, let K be the origin of... co-ordinates ; put Kn = x, wM = y, and SI, the radius of the generating circle, = a. Then we have, Mft = M?tt -f mn (1).
And Mm = QI = AI AQ (2).
Now from the mode of generation, we have, AQ = arc MQ =. Arc 7?? I; and AI = semi-circumference IniK. Hence (2) becomes, M??z = ImK arc ml= arc Km, and, consequently, (1) becomes, y = arc Km + mn = arc Km -f ^/Kn X nl arc Km + 236 ANALYTICAL GEOMETRY. [CHAP. VI.
Now we have arc Km = a circular arc whose radius is a and ver sin x = a (an arc whose radius is unity and ver sin -) ; or, ~ l x introducing the notation, ver sin - to signify " the arc whose x versed sine is, " (3) may be written, y = a ver sin - + >/ 2ax a 2 (4) for the equation of the cycloid.
What to read after An Elementary Treatise On Analytical Geometry?
You can find similar books in the "Read Also" column, or choose other free books by Jean Baptiste Biot to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
This is the curve which a nail in the rim of a carriage-wheel describes in the air during the motion of the carriage on a level road. The curve derives its name from two Greek words signifying "circle-formed. " The line AL over which the generating circle passes in a single revolution is called the base of the cycloid, and if I be the middle point of AL, the point K is called the vertex, and the line KI the altitude or axis of the curve. To find its equation, Fig. 14.
CL, let K be the origin of... co-ordinates ; put Kn = x, wM = y, and SI, the radius of the generating circle, = a. Then we have, Mft = M?tt -f mn (1).
And Mm = QI = AI AQ (2).
Now from the mode of generation, we have, AQ = arc MQ =. Arc 7?? I; and AI = semi-circumference IniK. Hence (2) becomes, M??z = ImK arc ml= arc Km, and, consequently, (1) becomes, y = arc Km + mn = arc Km -f ^/Kn X nl arc Km + 236 ANALYTICAL GEOMETRY. [CHAP. VI.
Now we have arc Km = a circular arc whose radius is a and ver sin x = a (an arc whose radius is unity and ver sin -) ; or, ~ l x introducing the notation, ver sin - to signify " the arc whose x versed sine is, " (3) may be written, y = a ver sin - + >/ 2ax a 2 (4) for the equation of the cycloid.
What to read after An Elementary Treatise On Analytical Geometry?
You can find similar books in the "Read Also" column, or choose other free books by Jean Baptiste Biot to read onlineMoreLess
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