ALGEBRAIC POINT FUNCTIONS IN N-SPACE. 403 



as before, a set of n vectors ai, a 2 , • • • a„. Consider for a moment the 

 case K = 2. A symmetrical dyadic may be written as a sum of terms 

 of the form aa where a is a vector. Let there be a set of constants 

 C\, Co, ■ • • c n not all zero. We define the equation 



ciaiai + c 2 a2a 2 + • • • + c n a n a„ = (12) 



to be equivalent to the set of n scalar equations obtained by selecting 

 pairs of corresponding components in all possible ways: — 



Ciauda + c 2 a 2 ia2j + • • • + c n a n ia n j = (13) 



where the subscripts i and j may have any values from 1 to N. These 

 equations are linear and homogeneous in the c's; hence the determi- 

 nant whose elements are audhj must vanish if (12) holds true. A 

 glance at (11) shows that this determinant is the same as Co, the minor 

 of F(x),— by interchanging rows and columns. Therefore the neces- 

 sary and sufficient condition for the existence of a dyadic equation of 

 the form (12) is that the n vectors which enter in that equation should 

 lie on a quadric cone. 



A symmetrical /v-ad should properly be written 



aaa • • • to K factors (14) 



but, when no ambiguity is brought about, we may write it as a K . We 

 in general define the A'-adic equation 



da/+ c 2 a 2 * + • • • + c n B, n K = (1|5) 



to be equivalent to the system of n equations 



1f[c h ahiahi • • • to K factors] = (16) 



h 



where in each equation h runs from 1 to n. Passage from one equation 

 of the system to another is by varying each of the second subscripts 

 from 1 to N. By inspection of (4) it is evident that the eliminant of 

 the set of n equations is the same as Co. We thus have 



Theorem II. A set of n symmetrical K-ads are linearly related if 

 and only if their vector elements lie on a hypercone of order K. 



4. Special Forms of the General Identity: Method of 



Standard Sets. 



By assigning particular values to the vectors ai, a2, • • • a„ and by 

 selecting particular forms of polynomials F(a), a great number of 



