364 Locus of the Conies which pass through four given Points. 

 fessor Sylvester, the equation 



\ v/S -f fju\/B + va/C + ttVD = 



(where X, yit, v, 7T are constants) is in general a curve of the 

 order 8; but the ratios X-.fi'.v.ir may be so determined that 

 the order of the curve in question shall be = 6 ; the resulting 

 curve of the order 6 is (not one of a group of curves, but the very 

 curve) the locus of the foci of the conies through the four points. 

 And the determination of the ratios X : \x : v : it is in fact quite 

 simple ; for writing 



k={x-af+(y-a l f 

 = p 2 — 2 (ax + a x y) + &c. 

 (hV=^+2/ 2 ), 



and therefore 



v /A=p- a ^±M+&c. ) 



with similar values for \/B, \/C, \/D, it is easy to see that, 

 considering \, /x, v, ir as standing for ■ + X, ±fi, ±y, + it re- 

 spectively, the conditions for the reduction to the order 6 are 



X +IJL +V +7T =0, 



\a + fxb -\-vc -\-nrd =0, 

 \a x + fjbb l -\- vc x + 7rd x = 0, 

 and hence that the required equation of the curve of foci is 



2{ 



1, 1, 



i 



b, c, 



d 



h> d i> 



*. 



v/(*-«) s +(y-«,)*}=0. 



or, as this may also be written, 



where (B, C, D), &c. are the areas of the triangles B, C, D, &c. 

 I remark, in conclusion, that the number of conditions to be 

 satisfied in order that a curve may have for double points two 

 given points I, J, may have besides six double points, and may 

 have for double tangents eight givenlines,is (3 + 3+6 + 16 = )28; 

 the number of constants contained in the general equation of the 

 order 6 is = 27. The conditions that a curve of the order 6 shall 

 have for double points two given points I, J, shall besides have six 

 double points, and shall have for double tangents four given lines 

 through I and four given lines through J, are more than sufficient 

 for the determination of the sextic curve ; and the existence of a 

 sextic curve satisfying these conditions is therefore a theorem. 



