THE LANCASHIBE GEOMETEBS AND THEIR WBITIKGS. 



151 



Porismatic Construction. By Mr. John Kay. 



" " Let G Q r R P be the circle,* and A, B, the points given 

 by position; through the centre O draw A C R«, making 

 R/j = 2ACl, and divide A Q in S, so that SQ.Qre^AQ"; 

 draw S Z perpendicular to A n, and S Z will be one of the 

 lines. In the same manner draw B P O w, making »« r = 2 B P, 

 and divide B P in y, so that y P — P w = B P", and draw the 

 perpendicular y to, which will be the other line." (Math. 

 Comp.j Ques. 570.) The construction is similarly given by 

 Messrs. Butterworth and Simpson ; the latter of whom shews 

 that the Porism is true for any number of points and lines. 



PORISM. 



Proposed by " P. P.," Mr. John Lowry. 



"Three circles, having their centres in the same straight 



line, being given by position, two straight lines may be found 



which will also be given by position, such, that if from any 



point in the circumference of one of the circles tangents be 



drawn to the other two, and perpendiculars be demitted from 



the same point upon the lines to be found, the rectangle of 



the tangents shall be a mean proportional between the sum of 



the squares of the two perpendiculars and a certain given 



space." 



\ 

 Porismatic Construction. By Mr. William Simpson. 



, " Let O c, F/, be the radii of the circles, to which tangents 

 are to be drawn^ A S that of the other, A S B its diameter. 

 From S, towards O and F, apply S c?, S c, such that Sd . SO 

 = S c . S F = S A». Bisect c? O and c F at A and g, from 

 which points towards S apply h G and g H, such that 2 O S . 

 A G = O c% and 2 S F . ^ H = F/". On G H as a diameter 

 let a circle be described to cut perpendiculars erected at A 

 and B, at C and E, and D and F. Join D C and FE to meet 

 * The Figures to these Constractioas may readily be supplied by the reader. 



