206 MATHEMATICS: J. L. SYNGE Proc. N. A. 
From (2) we find 
-■■■■■--l[:]-m-'''{[:m-[:i;]y^^> 
Since the presence of just one 3 among the indices makes the tensor- 
component vanish, the surviving independent members of this class are 
^31,13, Gsi,2S , G32,2Z , ^21,12 1 
we calculate them from (5) : — 
r,,,3 = i - 1 P-^ ~ (p!!y = - I sinH. ^ (6) 
\OxiJ 2 
2dxi 4 c>xi 4 Y>xi J 2 1-k/xi 
^ _ 1 C)^g33 1 ^22 ^^33 ^^22 1 ^33 (>gZ3 (>gZ3 _ ^ 
Cz31,23 - - — — — - g — ~ "a ^ ^ ' A ^ 
2 OX1OX2 4 0x2 0x1 4 0x2 0x1 
2 b^t;2 4 dxi bxi 4 \c)^C2 / 
r7 = I _ 1 pii ^ ^1 _ ^ ^22 = _ 1 ^Ai 
(7) 
2c>xi 4 bxi djti 4 / 2 1-^^1 
The complete list of surviving components, derivable from (4), (6) and 
(7), is as follows 
[ 6*21, 21 
6'i2,21 
[ 6^12,12 
6*21,12 
of type 6^5152,5,1 
"1 GziyZi 
6*13, 31 ; 
of type 6^5152,532 
j 6^32,32 
6'23,32 ; 
[ G ^1,4,1 
Gu,ii 
1 6^42,42 
6"24,42 
[ Gis,is 
631,13 
[ 6^14,14 
6-41,14 
of type Gs,s2,sz3 
\ <J23,23 
6^32,23 ; 
of type 6^5152,534 
\ 6^24,24 
6^42,24 . 
[ 6^43,43 
6'34,43 
[ 6^34,34 
6*43,34 
The equations 
of the principal directions of Type IV are 
Ogstdxt = 
Gi,t,,HtdoCi (5 = 1, 
2, 3, 4); 
(8) 
these become 
for 5 = 1, ^gnJxi = 2gii [fe22^2i,i2)2 + (g^^6^3i,i3)2 + (g^^6^4i,i4)2] dxu 
for 5 = 2, ^g22^i:r2 = 2g22 [(gll6'i2,2l)^ + (g^^6^32,23)2 + (g^^6^42,24) ^ 1 dX2\ 
for 5 = 3, Bgzzdxz = 2g'' {(g''Gis,ny + (g'26^23,32)2 + (g^^6'43,34)21 c?:^3 : 
for 5 = 4, Bgudx, = 2g44 [(gll6"i4,4l)2 + {g''G24,A2r + (g^^6'34,43)'l dX4. 
On substitution we obtain 
^2 ^2 
ddxi = S dxi, 6dx2 = 3 dx2, 
X^ Xi 
J^2 J^2 
Bdxz = 3 -g dxz, BdxA = 3 -g- dx^. 
X\ Xi 
