34) The Three Sections, Tangencies, 



circle OHA cuts MN in real points. Hence, it is necessary 

 to define tlie positive limiting values of i so as to be enabled 

 to know, a priori, wlien tlie corresponding points C are real, 

 &c. This is done by drawing the two straight lines PO'H' to 

 cut the circle PGQ and line ATI in O' and H' such that the 

 two circles AO'II' shall touch MN (these circles can be easily 

 described, since the point X, in which AG cuts the circle PGQ, 

 is common to all circles AHO. For the angle XG right to 

 O = PG right to O = HA right to O = XA right to O) ; for 

 then the limits are ^^' and ^^^' ; and tl^se limits are 

 evidently such that according as any value of j, lies outside 

 them, or is equal to one of them, or is comprehended between 

 them, so will the corresponding points C be real and distinct, 

 real and coincident, or imaginary. 



When the segments PS and QE have no part in common, 

 it is evident A and G are on opposite sides of MN, and that 

 it is only when the given ratio is negative that the points 

 OHIA can be on the same side of MN, and .-. only that the 

 points C can be imaginary. Hence, in this case, it is neces- 

 sary to define the negative limiting values of n- These 

 limiting values 1^" §xS'^ ^^'^ obviously found in the 

 same manner as in last case, and like remarks as to their 

 nature apply. 



When Q. coincides with S, it is evident the points C are 

 always real, and that one of them is coincident with QS. 



When P coincides with Pt, it is evident the points C are 

 real, and one of them in PR. 



When R and S are coincident, and P and Q are distinct, 

 then AR and AS are coincident, and G is at infinity, and .*. 

 O is in the straight line PQ; moreover, H coincides with 

 KSj .". the points C are real, and one of them coincident 

 with E. 



When P and Q, are coincident, and E, and S distinct; 

 then G coincides with P and O ; but although the triangle 

 POQ is infinitely small, it is known in species, and therefore 

 POH is known in position, and hence the circle OHA. The 

 points C are real, and one of them coincident with PQ. 



Porismatic Relations of the Data. 



If R and S be coincident, and that we have P and Q also 

 coincident, then as we may have conceived PQ and RS to 



