An elementary treatise on curve tracing . e. r INTERSECTION AT AN INFINITE DISTANCE 73 PLATEequation in x or y, whatever be the value of a. The recti- iv. linear asymptote which corresponds to the tangent to a branch, must then be found by the condition that one more root has an infinite value ; and this condition will determine the r parallel asymptotes. 103. Take as an example the curve of the third degree,whose equation is x(y-x)2-b2y = 0. Any straight line parallel to y = x has an equationy = x-{-OL and meets the curve where 0 . X^-^0 . X^ + (OL^-¥-)x-hOL = 0, so that there are two in
An elementary treatise on curve tracing . e. r INTERSECTION AT AN INFINITE DISTANCE 73 PLATEequation in x or y, whatever be the value of a. The recti- iv. linear asymptote which corresponds to the tangent to a branch, must then be found by the condition that one more root has an infinite value ; and this condition will determine the r parallel asymptotes. 103. Take as an example the curve of the third degree,whose equation is x(y-x)2-b2y = 0. Any straight line parallel to y = x has an equationy = x-{-OL and meets the curve where 0 . X^-^0 . X^ + (OL^-¥-)x-hOL = 0, so that there are two infinite values of x for all lines parallelto y = x. In order to obtain the nearest possible stiaightline to the curve at an infinite distance, we must makeoc^ = b^, so that either of the straight lines, y = x±h, meetsthe curve at three points at an infinite x = 0, the values of y are given by + = 0,which shews the axis of y as an asymptote. Near the origin, x^ — bhj = 0, ^V^ ; also the curve issymmetrical in oppos
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