338 Mr. Stubbs on a new Method in Geometty. 



low precipitates. When the lead salts were decomposed by 

 sulphuretted hydrogen, I obtained an amorphous acid sub- 

 stance of a bright yellow colour, which was soluble in water, 

 alcohol and aether, but which did not appear to be crystal- 

 lizable. 



XLII. On the application of a new Method to the Geometry 

 of Curves and Curve Surfaces. By J. W. Stubbs, B.A., 

 Trinity College, Dublin. 



To the Editors of the Philosophical Magazine and Journal. 

 Gentlemen, 



f HAD the honour of reading a paper before the Philoso- 

 -* phical Society of Dublin, on a new Geometrical principle, 

 which as far as I am aware has hitherto escaped the notice of 

 mathematicians. May I ask of you the favour of inserting 

 it in your valuable Journal? 



The principle consists in taking the inverse of curves and 

 surfaces, by meansof which we readily find conjugate properties 

 to those possessed by every known curve and surface, the dis- 

 cussion of many of which would be impossible by the ordinary 

 methods. If in the plane of a curve we take any point as a 

 pole and produce the radius vector, so that the rectangle 

 under radius vector to the original curve and the whole pro- 

 duced radius be constant or equal to k% we may call the locus 

 of the extremity of this produced line the inverse curve to the 

 one from which it is produced, and the extremity of the pro- 

 duced radius the inverse point to the extremity of the origi- 

 nal : as an example, the cardioide is the inverse of the para- 

 bola, the focus being the pole ; the lemniscata in the inverse 

 of the equilateral hyperbola. The inverse of a right line is a 

 circle, except when the pole is on the right line, when it is'a 

 right line. The inverse of a circle is a circle wherever the 

 pole is situated, except it be on the circumference, when it be- 

 comes a line perpendicular to the diameter through the pole. 



To draw a tangent to the inverse curve at the y ( . 

 inverse point to a given point on the direct or gene- 

 rating curve, join the points, and on the joining line 

 describe an isosceles triangle, one of whose sides is 

 the tangent to the direct curve. The other will 

 be the tangent to the inverse, as is seen by taking / 



two consecutive radii; from the property by which / 



it is generated the quadrilateral A M B C is cir- a 1m> 

 cumscribable by a circle; hence the angle A M C / 



equals the angle T B A, but in the limit the lines ro 



A M and B C become tangents : this is also clear 



'o' 



3^7 



