HBV. T. P. KIEKMAN ON LINEAR CONSTEUCTIONS. 2$3 



Generally, if n be not greater than the number of con- 

 stants m the general equation of the N* degree, the expres- 

 sion 



• bo •'b'c' a"b"o" «, b, e, o o o 



2 + xyz . «yz . xyz x y z , x y zznO 



000 111 S28 n-I n>l n-1 n n n 



is the equation to a surface of the N*^ degree passing 

 through the n points x^y-^z^^ XiViSn &c., where the indices ahc 

 are any positive numbers, zero included, different or alike, 

 as are also a! h' c', &c. ; provided first, that at least one set of 

 three has a sum a -\-b -if c -^l^i secondly, that no set 

 (except the last) be all three zeros ; thirdly, that no set 

 has a sum greater than N ; fourthly, that in no pair of sets 

 a'h'c', and aj),c,f we have at once a' zz a,, b' = b,, and c' zzc,; 

 and, lastly, that the indices are not reduced to a a^ ... alon* 

 — to 6 5^ ... alone — or to c c^ ... alone. 



In the same manner, we can write out at once the equa- 

 tion to a locus of any degree for a geometry of four dimen- 

 sions, the defining points being (x^ r/i Zi i/7,), (xs 2/2 ^2 W2), &c. ; 

 or with equal facility for a geometry of any number of 

 dimensions. 



Thus, for example, 2 + iK^ . y, . Ij ::r: 0, is a parabola having 

 for diameter the axis of y, and passing through the points 

 1 and 2 ; 2 ^j a^o yo • ^i ^^ ^ is an hyperbola through 1 , whose 

 asymptotes are the axes ; 2 + Xq 2/o • ^''i • 3/2 • I3 is an hyperbola 

 through 1, 2, and 3, whose asymptotes are parallel to the 

 axes; 2 ± (4+2/S) • a?i • ^2 • I3 ==0 and :s±(afo±fo).li=zO 

 are conies referred to equal conjugate diameters which are 

 parallel to the axes; these curves are either circles or 

 equilateral hyperbolas, if the axes chosen are rectangular. 

 The additional terms are in every case to be formed by 

 permutation of the sub-indices 



So far as I am aware, this is a view of equations to geo- 

 metrical loci which has not been given before. I know that 

 the shape which the results of elimination must assume, is 

 no secret to analysts; but has this simple mode of stating 



