80 The Three Sections, Tangencies, 



the second line NN ; through two given points Bj C, to draw 

 two straight lines BI^ CI, intersecting each other on LL, the 

 third given line, so that E and F being the respective points in 

 which BI and CI cut MM and NN^ we shall have PE to QE 

 in the given ratio of m to n. (The magnitude of j^ is not only 

 considered known, but its sign also — the directions on MM and 

 NN being considered.) 



ANALYSIS. 



From porisms in tlie Transactions for 1859, we know that 

 if we draw QA making tke angle QN right to A = P M 

 right to B, and that we make QA to PB as QF to PE or as 

 m : n, then Avill BE and AE intersect in a point O in the 

 circumference of the circle described through A, B, and the 

 point B of intersection of PB and QA. 



And from porism 8 in the Transactions for 1859, Ave know 

 that if we draw CG and CD parallels to NN and LL to cut 

 LL and NN in G and D, then wiU GI.DE be equal to the 

 known magnitude GC.DC. Therefore, since the points B and 

 A are known^ and also the circle ABR^ it follows (by problem 

 second, Transactions for 1859) we can draAv the straight lines 

 BOI and AOF, and therefore also GIF through I where BO 

 cuts LL. 



COMPOSITION. 



Through C draw CG and CD parallels respectively to NN 

 and LL, to cut LL and NN in G and D ; draw QA making 

 the angle QN right to A equal to PM right to B, and make 

 QA : PB : : n : m, describe a circle through A, B, and II 

 the intersection of PB and QA ; find a point O in the circlets 

 circumference (by problem second^ Transactions for 1859) 

 such that I and F being the points in which BO and AO cut 

 LL and NN, we shall have GI.DF = GC.DC; draw EC to 

 cut LL, then will BI and EC be answerable lines. 



For let E be the point inAvhich BI cuts MM. By porisms 

 (in Transactions for 1859) we haA^e the point I on the line 

 EC ; and PE : QF : : m : /i. 



PORISMATIC RELATIONS OF DATA. 



Before entering on the investigation of the porismatic 

 relations^ it may be as well to observe that it is evident that 



