88 
must, however, be noticed that Vp£ will not move with the 
fluid. 
Taking § as before to be a vector moving with the fluid 
and writing S = hvfa + 4v </> 2 4 hvfo we get 
V t S = DA- V 0 i + &o. 4 VpS 4 S3v<r by (1) = - (SSv)*. 
.’. the displacement at the extremity of 8 is 
-"save- = i VpS 4 J(DA.' V01 + DAv^a + DAvfc) 
Of these terms on the right the first denotes rotation about 
an axis £ ; for the second we will write JS 1 . Let € be a 
second vector moving with the fluid, and e 1 be the vector 
corresponding to S 1 . Then the condition that the part of 
the strain corresponding to 8 1 may be pure is that 8 1 may 
be self conjugate* or that SeS^SSe 1 . 
We have S 1 + VpS 4 2St)v<r = 0 
e 1 4 Vpe 4 2S£Vo > = 0. 
S eS 1 - S e 2 S = S3Vpe - S eVpS 4 2S(3SeV " eS3v)* 
= 2S Spe 4 2S Sep = 0. 
There are generally three principal axes of strain which 
are at right angles ; from the equations of condition which 
are V8S 1 = 0 we get for these axes 
DA DA DA _ D^ 
h ~ir k 1 y 
If 8 and e represent two of these principal axes 
( 10 ) 
D t S Se = SD t Se 4 S3D*e = - SeS3 V <r - S3Se V <r = SSe 1 = SS'e 
and since s 1 is parallel to s, and S8e = 0, the principal axes 
remain at right angles, and therefore remain principal 
axes. 
As this is important, we will give another proof of it:— 
S may be written l (^v 0 1 4 k 2 v0 2 4- zc 3 v0 3 ) where 
D^ = 0 by (10), and S 1 may be written 
D t l {'nV^i 4- /vW02 4 
For shortness write S = l£, S 1 = D t l.£ 
;.d,W=vd^ + vsda 
= v { (Q t vc+ ®,4)D t U - (DH.'C + 
= 0 . 
# Hamilton’s Elements of Quaternions, III. 2—6, and passim* 
