1919-20.] An Identical Relation connecting Seven Vectors. 129 
XIV— An Identical Relation connecting Seven Vectors. By F. L. 
Hitchcock, Massachusetts Institute of Technology, Cambridge, 
Mass. Communicated by The General Secretary. 
(MS. received April 23, 1920. Read June 7, 1920.) 
1. Utility of Vector Identities. 
Among the most useful results of quaternion or vectorial algebra we may 
count the identities and transformations worked out by Hamilton and 
Tait and their all too few followers. It is not merely that a vector 
identity is equivalent to three scalar identities — a fact which aids us 
greatly to condense our calculations in respect to bulk. Yet more 
important is the greater fruitfulness of a vectorial relation in giving 
rise to derived relations which we would be less likely to perceive from 
a purely scalar analysis. 
In this paper I propose, first, to develop an identical equation satisfied 
by any seven vectors and also satisfied by an arbitrary quadratic function ; 
second, to exemplify the methods by which new identities can be derived 
from this (or from any) vector identity. 
2. Fundamental Identities. 
Hamilton and Tait make constant use of certain fundamental relations 
connecting three and four vectors. There are also, among the examples 
in Tait’s Quaternions and in Kelland and Tait’s Introduction to 
Quaternions , a considerable number of identities connecting five or 
more vectors. Most frequently used are the two following : 
pSa/3y = a&ftyp + /3Syap + ySa/3p . . . (1) 
which expresses any fourth vector p in terms of its components along 
three other vectors a, /3, y ; and 
Y aV (3y — ySa/3 — /3Say ..... (2) 
which is of constant utility. 
3. The Quadric Cone through Five Vectors. 
The present paper deals with certain expressions which are quadratic 
forms in the sense that they are homogeneous of degree two in various 
sets of quantities. We begin with Hamilton’s expression 
S . Y (Y a/3VSe) V ( Y fiyV ep)Y (Y ySY pa) ... (3) 
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