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# An elementary treatise on elliptic functions by Arthur Cayley

By Arthur Cayley

This quantity is made from electronic pictures from the Cornell college Library historic arithmetic Monographs assortment.

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Extra resources for An elementary treatise on elliptic functions

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4-11. But Jm= u2 - u2 AB = U and AC = = /,. 2, lines AD and BC are perpendicular. tY D(u + U. w ) Fig. 2. Assume that 9, and 9, are perpendicular nonvertical lines of respective slopes m, and m, . We shall show that m1m2 = - 1. Let 14: be the line through the origin 0 and parallel to 9,, and let 14: be the line through the origin and parallel to 14, [see Fig. 4-12(a)]. 1). Also 14: is perpendicular to 9;since 9, is perpendicular to 14,. Let R be the point on 9fwith x-coordinate 1, and let Q be the point on 9 3 with x-coordinate 1 [see Fig.

Let Q have coordinates (U, U), and let B be the intersection point of 9 and the line PQ (see Fig. 6-9). 2, b-u=u-a b+a=v+u v+u=a+b tY I Fig. 6 9 1 45 SYMMETRY CHAP. 61 To solve (I) and (2) simultaneously for U and U, first add the two equations, yielding 20 = 2 4 or U = a Then subtract (I) from (2), yielding 2u = 2b,or U = b. Thus, Q has coordinates (b, U). If the graph of 3x2 + xy = 5 is reflected in the y-axis (that is, each point on the graph is replaced by the point symmetric to it with respect to the y-axis), find an equation of the new graph.

7-11(u)]. -A). tY AY -I 0 1 X Fig. 2 Find the domain and the range of the functionfdefined by The domain off consists of all x such that either - 1 < x < 0 or 0 I x < 1. This makes up the open interval ( - 1, 1). The range off is easily found from the graph in Fig. 7-1l(b), whose projection onto the y-axis is the half-open interval [0, 1). 3 Definef(x) as the greatest integer less than or equal to x; this value is usually denoted by [x]. Find the domain and the range, and draw the graph off: Since [x] is defined for all x, the domain is the set of all real numbers.