18260 



iiotutions of the Function ^' x. 



24o 



In the figure - = t, an.d 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 inte.rsection 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 = •?, "• 



