& 
INVESTIGATION of PORISMS. _ 185 
Tue analyfis of the Porifm, when thus generalized, is too 
long to be given here*. We muft 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 interfected by the others, there be placed weights 
proportional to the fquares of the fines of the angles of inter- 
feétion, 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 ftraight 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, fhould have 
a given ratio to the fquare of DR, it is plain, that the finding 
of the point Gin the Porifm, would render the conftruétion 
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- 
fition ; 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. 
Tue fame Porifms affift alfo in the folution of another 
problem. For if it were propofed to find D, fo that the fum 
Vou. IIL Aa of 
* Tars Porifm, in the café confidered above, vz. when there are three ftraight lines 
Biven in pofition, was communicated to me feveral years ago, without any analyfis or 
: _- demonftration, by Dr Trait, Prebendary of Lifburn in Ireland, who told me alfo, that 
he had met with it among fome of Dr Srmson’s papers, which had been put into his 
hands, at the time when the pofthumous works of that geometer were preparing for the 
prels. The application of it to the fecond of the problems, (§ 27.) was allo fuggefted 
by Dr Tran. 
