and Loci of Apollonius, S^c. 81 



as there are Uxo answerable pairs of lines BO^ AO (see 

 problem second, in Transactions for 1859), there are two pairs 

 of answerable lines BI, CI ; and that according as the lines 

 BO, AO are real and imaginary, so will BI CI be real or 

 imaginar3^ Moreover, it is evident that when ^ is not 

 restricted as to sign, there are fonr solutions to this problem, 

 two of which cannot be always real, &c. 



Now, it is eWdent (from porismatic theorems in Transactions 

 for 1859), that in the porismatic state of the data of the present 

 problem, we must have the angle DN right to A = angle GB 

 right to L, andDA.GB = DC.GC; and, moreover, a straight 

 line through A parallel to NN must cut BG in a point S on 

 the circumference of circle ABR. But (from ponsms in 

 Transactions for 1859) the angle OA right to B is equal 

 angle NN right to j\IM, and .•. angle SA right to B being 

 equal to this last angle, it follows that SGB must be parallel 

 to M]M. Again, for any finite value of ~, it is evident that 

 QC must pass through the point H in which BP cuts LL, 

 It is also evident the triangles BGC, CD A, are similar. And 

 if T be the point in which DA cuts the circle ABOSR, we 

 have the angle BS right to T = AS right to T = DN right 

 to A := GB right to L; and therefore BT is parallel to 

 LL. 



Hence, we see that in the porismatic state QC and PB 

 intersect on the line LL, and that BG and CG parallels 

 respectively to MM and NN intersect on LL, and that the 

 ratio ^ is equal to the ratio of the segments PU and QD, 

 made on MM and NN by straight lines BU and CD parallel 

 toLL. 



And we deduce the following important porisms : — 



PORISM. 



Given two j)oints B, C, and two straight lines MM, NN, in 

 position, and the point P in MM, atid the point Q in NN; a 

 straight line LL can be found, such that I being any point in^ 

 it, and E and F tJie respective points in tvhich IB and IC cut 

 MM and NN, we shall have PE to QF in a constant deter- 

 minable ratio. 



For BG and CG parallels to MM and NN, give us one 

 point G in the required line ; and QC and BP give us another 

 point in the required line by their intersection : therefore the 



