and Loci of Apollonius, ^c. 77 



being done^ it follows that the a.b for the second enmiciation 

 is negative, and .-. according as AO or ^ AB is less than, 

 equal to, or greater than A/ or '^ or -^^ so will the locus be 

 real, real and infinitely small, or imaginary ; or which amounts 

 to the same thing— according as ~ AB^ is less than, equal to, 

 or greater than (f.h, so will the locus be real, real and 

 infinitely small, or imaginary. 



Again, Avhen -^ is positive (and therefore O is in a production 

 of AB) according as g.h in the first of these two enunciations 

 is negative or positive, we can take m and n, both positive or 

 both negative. Doing this makes the corresponding a.b always 

 positive, and gives the point/ which does not coincide Avitli 

 B always in the production of AB through A. Hence we 

 see that the locus is always real in this case when m. is nume- 

 rically gi'cater than n. But when m is nmnerically less than 

 n, it is evident that according as AO or (,-;^JAB is longer 

 than /A (or ^ or ,7^), or equal to it, or less than it, 

 so will the locus be real — real and infinitely small— or 

 imaginaiy; or, which amounts to the same thing, according 

 as (;,777()-^^^ is in extent greater than — equal to — or less 

 than g.h) so will the locus be real — real and infinitely small 

 — or imaginary. 



NOTES. 



In respect to the general loci problem just investigated, it 

 may be remarked that if S be the point in which CD cuts 

 AB, then Avill AS : BS : : AC.PD : BD.PC. And since 

 this last ratio is evidentlv compounded of the knoAvn ratios, 

 AC.AP to BD.BP, and PD.PB to PC.PA, it follows that the 

 ])oint S is known, as also the magnitude of the rectangle 

 SC.SD. 



And from this it is evident the problem may have had its 

 origin in the folloAving 



GENUINE* ANCIENT PORISM. 



If the sides ab, be, cd, de, ef, fg, of a pohjcjon aljcdefg in- 



* The anciant xioriams were formal investigations for theorems, and for 

 the relative states of the data of problems whicli might c.avise the solu- 

 tions to be immmerable. The modern pori^ms are but theorems originally 

 derived from porismatic researches. 



