1888-89.] Prof. Tait on the Relation among Four Vectors. 89 
SaS — #Saa + ySa/5 +5:Say , 
S/58- xSfia + ySfiP + zSPy , 
Sy8 =■ a?Sya + ?/Sy/5 + sSyy , 
S88 = xSSa + yS8/5 + zS8y . 
Hence, at once, a determinant of the 4th order. 
If we note that each term, as S/5y for instance, can he written 
2 A 
either as %(P 2 + y 2 - P - y ) or as - T/5Tycos/5y, we see that the 
determinant may he written either in Dr Muir’s form or as 
0 = 
1 
A 
COS aP 
A 
COS ay 
A 
COSa8 
A. 
COSpa 
1 
A 
COS Py 
cos/58 
A 
COSya 
A 
COS yp 
1 
A 
COSy8 
cos 8a 
cos 8/2 
A 
cos8y 
1 
which is the relation among the sides and the diagonals of a 
spherical quadrilateral. The method above can, of course, be 
extended to any number of points. One additional point introduces 
three new scalars to be eliminated, and six new scalar equations for 
the purpose. 
{Addition — Read March 18.) 
If we operate, as above, with any other four vectors, we have 
Scqa 
Sfta 
S 7l a 
SS-ja 
S <lJ3 Scqy ScqS 
S&/2 Sfty S&S 
Syi P s yr/ Sy x 8 
SS ,/3 SSj-y SSjS 
and the tensors are again factors of rows or columns. Thus, if 
ABCD, abed , be any two spherical quadrilaterals 
cos Act 
cos A b 
cos Ac 
cos Ad 
= 0 
cos Bet 
cos Bfr 
cos Be 
cos B d 
cos C a 
cosCfr 
cos Cc 
cos C d 
cos Det 
cosDfr 
cos Dc 
cos D d 
This has many curious particular forms ; one, of course, being the 
former result, when the two quadrilaterals coincide. Another is 
