and Loci of Apollonius, S^c. 35 



have had any peculiar relation, such as a constant ratio, &c., 

 during tlieir diminution, it is evident we may suppose G any- 

 where whatever in the line PQG. And it is clear that for all 

 values of j. other than unit)-, the point O must coincide Avith 

 PQ, and one point C be coincident Avitli PQO, and the other 

 point C with RSH. But for the value of -^ = unity, the 

 point O may be anywhere in the circle GQP (which touches 

 ^I^NI in PQ), and .•. a point C may be anywhere in MM — 

 the state of the data being " porismatic." And it is evident 

 that when P coincides with R and S -with Q, and that -jr = 

 unity, then, also, vnW. the problem be " porismatic." 



Peculiar Case. 



If in MN we suppose SR. and RU equals respectively to / 

 and k ; then we have PC.SC : QC.RC : : SR : RU. 



Now, if we suppose R and U to remain fixed, and S to 

 become greater and greater in distance from the fixed points, 

 until it vanishes at infinity, then for points C at a finite 

 distance, we shall have SR = SC, and hence 



PC.RU = QC.RC. 



This case of the problem can be expressed as follows : — 

 " Given three points, P, Q, R, in a straight line, to find another 

 C in the same, such that K being a line of given length (and 

 knoivn sign in respect to the directions on ^IN^), ive shall have 

 PC.K = QC.RC.^^ 



The solution may evidently be worded thus : — In MN 

 make RU = K ; draw RA (not in ]MN) equal RU ; draw 

 PO and QO, making the angles PO right to Q, and QO right 

 to P equal respectively to AR right to U, and UR right to A ; 

 produce PO to cut RA. in II ; describe the circle OHA, and 

 it Avill cut jNIN in the required points C, C. 



