84 The Three Sections, Tatigencies, 



If the ratio -^ be not restricted in sign^ it is evident there 

 are two positions for QA equally distant from the intersection 

 of KK and NN, and therefore also two answerable lines LL 

 corresponding to the points Q, and Q. 



PROBLEM II. 



Given three straight lines MM^ NN^ and LL in position; 

 and the points P a7id Q, in the two first lines — one in each; 

 through tivo given points B^ C, to draiv straight lines BI. CI, 

 intersecting each other on the third given line, so that E and F 

 being the points in which they cut MM and NN (the first and 

 second lines) ive shall have PE.QF equal a given magnitude 

 m.nfthe sign o/m.n is supposed known, ^c.J. 



ANALYSIS. 



From porism 10 in the Transactions for 1859, it is obvious 

 that if we di'aw QA making the angle QN right to A = the 

 angle PB right to M, and QA.PB = QF.PE = m.n, and that 

 through A we draw a parallel to NN to cut PB in K, then 

 will FA and EB intersect in a point O in the circumference 

 of the circle ABK. 



And fi'om porism 8, in the Transactions for 1859, it is evi- 

 dent that by drawing CG and CD parallels respectively to 

 NN and LL, to cut LL and NN in G and D, we will have 

 GI.DF = GC.DC = a known magnitude. And since 

 through A and B, the lines AO and BO (intersecting in the 

 known circumference AKB), make GI.DF of a known mag- 

 nitude; therefore (by the second problem in the Transactions 

 for 1859) AO and BO are known in position, and hence the 

 points E and F, and line FCI, &c. 



REMARKS. 



The Composition and Discussion may be easily made if 

 thought necessary, but as my chief object is to develop the 

 porismatic relations of the involved data, I will at once pro- 

 ceed to do so. 



Porismatic Relations of Data. 



It is CAddent, from porisms in the Transactions for 1859, 

 that this problem will become porismatic when the angle GL 

 right to B = DA right to N, and that GB.DA = GC.DC. 

 And if K' be the point in which GB cuts the circle ABK, 



