DIFFEKENTTAI. INVARIANTS OF SPACE. 
;]27 
But 
Xj /9j + = 
.r, 
4>' 
100 5 
lOOl 
X, , X., 
^ 010 5 4’ 001 
4 010 ’ 4 001 
•^1 ’ ’ a’g 
qy"^,c^ + + cp",z, , +cR"..,, qV'^x. + + cR"x'3 
= L, 
and similarly for + yj, + y.O^, + zM. + ; lienee 
r’„ = L^- {((!>>", - ci>'Ay"^y + - cR>" )'2 + {cy',<p'\^ - d^yCR" .)'^] 
= JJ [(cR7 + CR'/ ^ _|_ ^ 
.oA/fyA/f'f -2^ 
= ^' idn’) [m') 
■ir„ 
Tills result is useful for the Identification of the invariant I. We have 
I = 
4100 > 4ow > 4mi 
4 '100 5 4'o\o > 4'ooi 
4"ioo^ 4"o\o^ 4"oo\ \ 
— ^l4]00 + ^24o\0 + ^s4oOl 
^ d<l, eJef,' cl<f>" . 
ch ch' dn" 
This expression can be 
modified so as to become skew symmetric in the three 
surfaces. Take a sphere, having its radius unity and its 
centre at the point; and let the directions • ds, els', ds", 
dn, dn', dn" cut the surface at S, S', S", N, N', N" 
Then SS'S" is the polar triangle of NN'N"; and 
N'N" =: n, N"N = n', NN' = fl". By a known pro¬ 
perty of such triangles, we have 
cos NS sin o = cos N'S' sin fl' = cos N"S" sin D." 
= (l — cos^ — cos" n' —cos" n"d- 2 cos n cos O' cos n")i 
= V, 
