An Introduction to Mathematics By a N Whitehead
An Introduction to Mathematics By a N Whitehead
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We then get the equation x 2 -\-y 2 4 =0. It is easy to prove that this is the equa- tion of a circle, whose centre is at the origin, and radius is 2 units of length. Now abh 2 becomes 1x1 O 2, that is, 1, and is therefore positive. Hence the circle is a particular case of an ellipse, as it ought to be. Genera- lising, the equation of any circle can be put into the form a(x 2 -\-y 2 } -\-2gcc -\-2fy+c=0. Hence abh 2 becomes a 2 0, that is, a 2, which is necessarily positive. Accordingly all cir...cles satisfy the condition for ellipses. The general form of the equation of a para- bola is so that the terms of the second degree, as CONIC SECTIONS 143 they are called, can be written as a perfect square. For squaring out, we get d z x 2 +2dexy +e 2 y 2 +2gx+2fy +c ; so that by comparison a=d 2, h=de, b=e z, and therefore ab h 2 =d 2 e 2 (de) 2 =0. Hence the necessary condition is automatically satis- fied. The equation 2xy 4=0, where a =b =g=/=0, h=l, c= 4, represents a hyper- bola. For the condition abh 2 becomes I 2, that is, 1, which is negative.
What to read after An Introduction to Mathematics By a N Whitehead?
You can find similar books in the "Read Also" column, or choose other free books by Whitehead, Alfred North, 1861-1947 to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
We then get the equation x 2 -\-y 2 4 =0. It is easy to prove that this is the equa- tion of a circle, whose centre is at the origin, and radius is 2 units of length. Now abh 2 becomes 1x1 O 2, that is, 1, and is therefore positive. Hence the circle is a particular case of an ellipse, as it ought to be. Genera- lising, the equation of any circle can be put into the form a(x 2 -\-y 2 } -\-2gcc -\-2fy+c=0. Hence abh 2 becomes a 2 0, that is, a 2, which is necessarily positive. Accordingly all cir...cles satisfy the condition for ellipses. The general form of the equation of a para- bola is so that the terms of the second degree, as CONIC SECTIONS 143 they are called, can be written as a perfect square. For squaring out, we get d z x 2 +2dexy +e 2 y 2 +2gx+2fy +c ; so that by comparison a=d 2, h=de, b=e z, and therefore ab h 2 =d 2 e 2 (de) 2 =0. Hence the necessary condition is automatically satis- fied. The equation 2xy 4=0, where a =b =g=/=0, h=l, c= 4, represents a hyper- bola. For the condition abh 2 becomes I 2, that is, 1, which is negative.
What to read after An Introduction to Mathematics By a N Whitehead?
You can find similar books in the "Read Also" column, or choose other free books by Whitehead, Alfred North, 1861-1947 to read onlineMoreLess
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