1888-89.] Rev. M. M. U. Wilkinson on Scalar Relations. 773 
On the Scalar Relations connecting Six Vectors. By the 
Rev. M. M. U. Wilkinson, Reepham Vicarage, near Nor- 
wich. Communicated by Professor Tait. 
(Read July 15, 1889.) 
A. Introduction. 
1. In the case of two Vectors, a, J3, the Scalars a 2 , /3 2 , Sa/? are 
connected hy no relation. In other words, the Tensors of two 
straight lines, and the angle between them, are three independent 
quantities. In this case every other Scalar Function of a and j3 
can he expressed in terms of a 2 , ft 2 , Sa/?. Thus, 
Sa/?a/?=2S 2 a/?-a 2 /? 2 . 
For convenience we shall call Scalars of the form - a 2 , Tensor 
Scalars, and Scalars of the form Sa /?, Primary Scalars. 
2. The introduction of a third Vector, y, introduces three fresh 
Scalars, as the Tensor of y, and its two inclinations to the Vectors 
a, /?. In this case we have six independent Scalars, in terms of 
which every other Scalar involving only the three Vectors can he 
expressed. Thus, a 2 , /? 2 , y 2 , Sa /?, S /?y, Sya, are independent Scalars. 
All other Scalar functions of a, /?, y, connect themselves with these 
by equations. Thus, 
S 2 a/?y = 
Say , Sa /? , a 2 
S/?y, /? 2 , Sa/? 
y' 2 , S/?y , Say 
. . ( 1 ) 
3. In general, if we have n Vectors, we have 3 (n - 1) independent 
Scalars, as each fresh Vector introduces three fresh Scalars, namely, 
its Tensor and its inclination to any two of the other Vectors. All 
other Scalars involving the n Vectors can he expressed in terms of 
the 3(?&-l) independent Scalars. Now, with n Vectors we have 
n Tensors, and n ^ n ~ ^ Primary Scalars, making Scalars 
I . Z 1 . z 
m 
all 
. As of these 3 (n - 1) are independent, it follows that ^ 
independent equations connect the Tensors and Primary Scalars. 
VOL. xvi. 13/2/90 3 d 
