. Annals of Philosophy. 18260 iiotutions of the Function ^' x. 24o. In the figure - = t, only those portions of the ordinates are drawn which termir;ate in P Q ; and being connected by- portions of lines paralhil to the abscissae, the whole forms an irregular polygon of« e(^ual angles, and 2 n sides equal by pairs, which is but another (^tfect of <f, as deducting equal portions from every ordinate an d abscissa. When N falls upon X or Y, this construction fails ; for X V being a tangent to th(i hyperbola, wherever the abscissae com- mence, the ordinates \vill continually approach the p
. Annals of Philosophy. 18260 iiotutions of the Function ^' x. 24o. In the figure - = t, only those portions of the ordinates are drawn which termir;ate in P Q ; and being connected by- portions of lines paralhil to the abscissae, the whole forms an irregular polygon of« e(^ual angles, and 2 n sides equal by pairs, which is but another (^tfect of <f, as deducting equal portions from every ordinate an d abscissa. When N falls upon X or Y, this construction fails ; for X V being a tangent to th(i hyperbola, wherever the abscissae com- mence, the ordinates \vill continually approach the point of con- tact, but never pass it. This corresponds to the case k = ii. It equally fails when N falls upon C. For the asymptotes being properly neither abscissae nor ordinates, and no others, in our solution, being connected with the point C, this is not a correct origin of abscissve, so that no transfer of such origin can be made. AccordiniJly the only real lines of abscissie are those which pass through the vertex of one of the iiyperbolas, and the origin is at their with the conjugate diameter, agre- ably to equation (30;), into which no circular function enters. Hence the ordinate and its abscissa are always of the same affection, both positive or both negative, which never happens in the general constiiiction, nor in equation (37); one or more changes of sign bei tig necessarily engaged in completing the circle of operations. This exception attaches to k = •?, "•. Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of these illustrations may not perfectly resemble the original London
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