R
i
ð Þ
j
0 i < j n
ð
Þ , and R i R
0
ð Þ
i ,
ð17:83Þ
where R
i
ð Þ
j is defined as a transformation R j described in reference to the coordinate
system i; R
0
ð Þ
i
means that R i is referred to the original coordinate system (i.e., the
fixed coordinate). Then we have
R
iÀ2
ð
Þ
iÀ1 R
iÀ1
ð
Þ
k
¼ R
iÀ2
ð
Þ
k
R
iÀ2
ð
Þ
iÀ1
k > i À 1
ð
Þ :
ð17:84Þ
Particularly, when i ¼ 3
R
1
ð Þ
2 R
2
ð Þ
k ¼ R
1
ð Þ
k R
1
ð Þ
2
k > 2
ð
Þ:
ð17:85Þ
For i ¼ 2 we have
R 1 R
1
ð Þ
k ¼ R k R 1 k > 1
ð
Þ:
ð17:86Þ
We define n time successive transformations on a moving coordinate system as f
R n
such that
f R n R 1 R
1
ð Þ
2 R
2
ð Þ
3 Á Á ÁR
nÀ3
ð
Þ
nÀ2 R
nÀ2
ð
Þ
nÀ1 R
nÀ1
ð
Þ
n
:
ð17:87Þ
Applying (17.84) on R
nÀ2
ð
Þ
nÀ1 R
nÀ1
ð
Þ
n
and rewriting (17.87), we have
f R n ¼ R 1 R
1
ð Þ
2 R
2
ð Þ
3 Á Á ÁR
nÀ3
ð
Þ
nÀ2 R
nÀ2
ð
Þ
n
R
nÀ2
ð
Þ
nÀ1 :
ð17:88Þ
Applying (17.84) again on R
nÀ3
ð
Þ
nÀ2 R
nÀ2
ð
Þ
n
, we get
f R n ¼ R 1 R
1
ð Þ
2 R
2
ð Þ
3 Á Á ÁR
nÀ3
ð
Þ
n
R
nÀ3
ð
Þ
nÀ2 R
nÀ2
ð
Þ
nÀ1 :
ð17:89Þ
Proceeding similarly, we have
O ⟶ I ⟶ II ⟶ III ⟶ ⋯
x
z
y
( ) ≡
( )
( )
( )
Fig. 17.17 Successive orthogonal transformations and relevant coordinate systems
670
17 Symmetry Groups