This historic book may have numerous typos and missing text.
Purchasers can download a free scanned copy of the original book
(without typos) from the publisher. Not indexed. Not illustrated.
1873 Excerpt: ...arc; the points of intersection will be points of
the curve. For, we have F'M--FM = 2a. It may also be constructed by
a continuous movement. Take a rule of sufficient length as F'L, and
fasten one end at the focus F'; at the other end of the rule fasten
one end of a string shorter than the rule by the transverse axis;
fasten the other end of the string at the other focus, F; press a
pencil against the string and rule; as the rule revolves about the
focus F', the point of the pencil will describe the branch AM. For,
we have F'L--2a = FM + ML, or F'L--ML-FM = 2a; henco F'M-FM = 2a.
By placing the end of the rule at F, the other branch may be
described. 122. By a reference to equations (1) and (2), Art.
(120), it, is seen that the distance from any point of the curve to
either focus is expressed rationally in terms of its abscissa. This
remarkable property of the foci is possessed by no other points in
the plane of the curve. For, if there is any other point, let its
co-ordinates be x' and y'; x and y denoting the co-ordinates of any
point of the curve. The square of the distance from x, y, to x',
y', Art. (17), is D = (-x'Y + (y-y'Y, or squaring x--x' and y--y',
and substituting for y its value --Va-A we have D = a b2 x2--2xx' +
x" + Vs =F 2y'--Vau _ x, yi a a 'y It is evident that the value for
D can not be rational, in terms of x, unless the term containing
the radical disappears. But this can not be unless y' = 0, that is,
the required point must be on the axis of X. Substituting this
value for y', I), after changing the order of the terms, becomes D2
= (6 + x'2)-2xx' + ffi" x. a% Now no value of x' can make this
expression a perfect square unless it makes the first and last
terms perfect squares, and twice the square root of their product
|Country of origin:
• Albert Ensign Church
||246 x 189 x 3mm (L x W x T)
||Paperback - Trade
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