/4 TJie Three Sections, Tangencies, 



" To find the locus of a point P such that the sum of the square 

 of its distance from a given point A and a given negative mag- 

 nitude a.h. shall be equal to the rectangle under its distance 

 VlLfrom a given straight line IK^ and a line 2.m of a given 

 magnitude whose sign is known in respect to the direction of the 

 2jerpendicular to the given line IK.'^ 



And since AO : BO : : AP.CP : BP.DP and that, in this 

 case, BO = BP, therefore AO.DP = AP.CP = jyV.m, and 

 hence AO = m. Therefore the solution for the problem just 

 enunciated may be worded as follows : — 



In AI perpendicular to IK find O such that AO = ^ 

 (twice m) =m; assume any point P', and in AP' find C such 

 that AP'.C'A = a.h.; di'aw P'K' perpendicular to IK, and 

 extend it on the other side of IK until K'D' = P'K' ; describe 

 the circle C'P'D' ; draw a tangent from O to this circle ; 

 from O as centre with this tangent as radius describe a circle, 

 and it will be the required locus. 



It is to be observed that the locus is real and finite, real and 

 infinitely small, or imaginary according as O is outside, on, or 

 inside the cii'cle C'P'D'. It is evident that when — a.b is greater 

 than AI^ the circle C'P'D' aoes not cut AI in real points / 

 and/, and therefore that the locus is real for aU real values of 

 m. When — a.b = AP the circle C'P'D' touches AI in I, 

 and the points / coincide in I ; and the locus is always real 

 for real values of m, and infinitely small when m = AI. 

 When a.b is less than AP the circle C'P'D' cuts AI in real 

 points / and / equidistant from I, and on the same side of 

 A as I j and it is e^ddent the locus is real for all values of m 

 having a different sign to the direction AI ; and, moreover, 

 for a value of m having like sign with direction AI, it is 

 evident that according as it is not comprehended between the 

 limits A/ and A/, or is equal one of them, or is comprehended 

 between them, so will the locus be real, real and infinitely 

 small, or imaginary. 



Fiirther, if we suppose a.b = zero, the limits A/ and A/ 

 are CAddently = zero and twice AI, and include the values of 

 m for which the locus is imaginary. 



When a.b is positive and c.d negative. 



Here it is evident the point C is on a different side of 

 AB with the points D', P', and that the circle P'C'D' cuts 

 AB, in one point between A and B, and in another in the 



