300 NOTES. 



and also every point in the diameter produced gives a line 

 cutting the circumference, and whose square is equal to the 

 rectangle of the segments of the diameter. 



So the data may be such as to render the solution im- 

 possible, and a change of these data making the solution 

 indeterminate, a porisni results. Thus, let it be required to 

 draw from a given point in the diameter produced a line cut- 

 ting the circumference, such that its square shall be equal to 

 the rectangle contained by the whole line and that portion of 

 it between the point and the ordinate to the point where the 

 circumference is cut; then there is no such point of the 

 diameter beyond the circle, because the square of the line 

 drawn to cut the circumference must always be less than the 

 rectangle under the segments of the diameter ; but/* being as 

 before = d* -J- rdx -j- ax, and being also = to (a + d) (d + x) 

 we have 



d* -\- 2 d x + ax = d 2 -^ dx -\- ax + ad, max = ad; 



and if d = 0, or the point is the extremity of the diameter, 

 / z = a x, and any line drawn to any point of the circumference 

 answers the conditions ; so that when the problem is impos- 

 sible, as well as when it admits of a determinate solution, a 

 change of the data making it indeterminate will give rise to a 

 Porism. 



Again : an ellipse and a point without it being given, and 

 a chord of the ellipse, let it be required to draw a straight 

 line from the given point cutting the ellipse and the chord, 

 so that it shall be divided by the ellipse and the chord in har- 

 monical proportion ; only one such line can be found, unless 

 in the case of the chord being so situated that the tangents 

 from the given point touch the ellipse at the extremities of 

 the chord ; and in that case every line drawn from the point, 

 and cutting the ellipse and the chord, is divided in har- 

 monica! proportion. Solving the problem algebraically; if 

 the equation to the ellipse be a 8 ?/ 8 + tf(x c) 8 = a*6 8 ; and 

 that to the given chord x = d y + n ; and that to the chord 



required x = my; and so m the quantity to be found we 

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