INVESTIGATION of PQRISMS. 185 



The analyfis of the Porifm, when thus generalized, is too 

 long to be given here *. We mufl not, however, omit to take 

 notice, that the point L, where the perpendicular from the point 

 to be found meets the line, from which the perpendiculars are 

 drawn to the reft, is in all cafes determined by the rule fuggeft- 

 ed in the corollary. (§ 24.) For if at the points in which the 

 faid line is interfered by the others, there be placed weights 

 proportional to the fquares of the lines of the angles of inter- 

 feclion, L will be the centre of gravity of thefe weights. 



27. These Porifms facilitate the folution of the general 

 problems from which they are derived. For if it were pro- 

 pofed, three flraight lines AB, AC, BC being given in pofition, 

 and alfo a point R, (fig. 6.) to find a point D in one of the 

 given lines AB, fuch, that the fum of the fquares of the per- 

 pendiculars drawn from D to the other two lines, mould have 

 a given ratio to the fquare of DR, it is plain, that the finding 

 of the point G in the Porifm, would render the conftruclion 

 eafy. For the fquares of RD and GD, having each given ratios 

 to the fum of the fquares of the perpendiculars drawn from D, 

 have a given ratio to one another. The ratio of the lines, RD 

 and GD themfelves, is therefore given, and the points R and G 

 being given, D is in the circumference of a circle given in po- 

 rtion ; and it is alfo in the ftraight line AB given in pofition ', 

 therefore it is given. The fame holds, whatever be the num- 

 ber of lines given in pofition. 



The fame Porifms aflift alfo in the folution of another 

 problem. For if it were propofed to find D, fo that the fum 



Vol. III. A a of 



* This Porifm, in the cafe considered above, viz. when there are three ftraight lines 

 given in pofition, was communicated to me feveral years ago, without any analyfis or 

 demonstration, by Dr Trail, Prebendary of Liiburn in Ireland, who told me alfo, that 

 he had met with it among fome of Dr Sim son's papers, which had been put into his 

 hands, at the time when the pofthumous works of that geometer were preparing for the 

 prei-. The application of it to the fecond of the problems, (§ 27.) was alio fuggefttd 

 by Dr Trail. 



