1888 - 89 .] Rev. M. M. U. Wilkinson on Scalar Relations. 777 
Whence, among many other expressions for W 2 , we have, 
w 2 = (B 5 + B 1 + B 2 ) 2 (B 5 +B 3 +B 4 ) 2 
+ (C 5 + C 1 +C 3 ) 2 (C 5 + C 2 + C 4 ) 2 
+ (E 5 + E 1 + E 4 ) 2 (E 5 + E 2 + E 3 ) 2 
- 2(C 5 + C 4 + C 3 )(C 5 + C 2 + C 4 )(E 5 + E, + E 4 )(E 5 + E 2 + E 3 ) 
- 2(E 5 + Ej + E 4 )(E 5 + E 2 + E 3 )(B 5 + B 4 + B 2 )(B 5 + B 3 + B 3 ) 
- 2(B 5 + B 1 + B 2 )(B 5 + B 3 + B 4 )(C 5 + C 4 + C 3 )(C 5 + C 2 + C 4 ) ; (15) 
we postpone for the present the consideration of the expansion of 
this, which will, of course, he symmetrical in form as well as in 
reality. 
10. From the known formula for 8 vectors, 
Sotot-j^ j 
S/toj , 
Sya x , 
S8a 4 
= 0; . 
• (16) 
Sa^j, 
mi. 
Syft. > 
Sayj , 
S/5 ri , 
s ryi » 
SSyi 
SaS 4 , 
m , 
SyS 15 
sas x 
putting a 4 = a, /?! = /?, “ft = c, S 4 = t, we find 
where 
Now, 
IV/3 2 + D 2 a 2 + Dg/3 2 + D 4 = 0 ; 
Di = 
Sye , SSe 
Sy£, 
Sy ; 
I>2 = 
0 , Spy, S/J8 j ; 
S fie , Sy€ , SSe 
SyC, SSC 
S0£SjS8ye-S0eSj88y£ = 
• - (O) 
• • ( 18 ) 
SPZSP&Syt - Bpt&PySSi - SfcSpSSyZ + S/?<S/?yS8£ ; 
So that 
D 2 = - (S/?£S/%e - S/?eS£Sy£) ; .... (19) 
So too, 
Do = — (Sa£SaSye - SaeSa3y£) ; .... (20) 
