Mr. Lubbock on some Problems in Analytical Geovietry. 87 



be transferred to polar coordinates by putting r a" for x" and 

 r b" for y, r being the distance of any point from the origin, 

 {Aa"'- + Ba"b" + CW^)r'' + {Da" + EW)r + F=0 



r -Da"-Eb' -N 



r = < ± {(i)' - 4 AF) a"^~ + 2 (Z)£ - 4 Bi^) a" 6" I 



L +{W-^cF)W% j 



2(^a"- + 5 a" 6" + C^>"2) " 



and since a"^ + W^ + 2 a" 6" cos xy =■ \ 



[-Da"-Eb" 

 r = J ±{ Z)--4^i^+ 2{DE-2BF-D--4JF) cos xy}c' 



[ +{E^-4^CF-D' + iAF}b'm 



'2 {A a"-' + Ba" b" + Cb"') 

 which value of r will be rational in terms of a" and J" if 

 ^3 _ 4 c-i^— Z>2 + 4 y4 i^ = 

 DE-2BF - (D' -4!AF) cos a;j/ = 

 putting for A, B, C, D, E and F their values above, it will be 

 found, after many reductions, that these equations may be put 

 in the form 



{a b-a bj 0'- - a'2) = {a" — b^) m' - n') 

 {jJb — a Uf (a! + |3'- cos x' y) = {if'—nf) {a- cos a^ y' — ab) 

 In these equations a! and /3' are measured from the center 

 of the curve in the direction of the axes x' and y : the equa- 

 tions to the same curve, found either by similar substitutions 

 for the quantities A, B, C, D, E, and F or by transforming 

 the last two equations, referred to coordinate axes coinciding 

 with the principal axes of the curve, are 



{a'^-b'^) («2-/32) + 2 {bb'-aa') a^ = (a'-'-b'^) (m'—fi^) 

 aa' («^-^') + (a'--b'^)cci3 = aa' {in^-n~) 

 making /3 = in these equations 



0L~ = m' — n^, cc = ± \/ (?«- — «-) 

 making a = 0, /3 = + v' {n^—nf) 



If 7K > n, the latter value of /3 is imaginary, and therefore 

 these curves cut one another in two points only, which are si- 

 tuated on the major axis at the distance + \/ iiv — n^ from the 

 centre. This result is entirely independent of the quantities 

 a, b, a', 6*, that is, of the direction of the axes x and y ; and 

 hence it is evident that there are only two points from which 

 the distance r to the curve is rational in terms of a' and b" . 

 These points are called the foci. 



Tlic 



