86 
Now V v o’ or p can be written SiH.-^ + 2 2 H^ + 2 3 H.!jf . 
D*p = D*logH.p- (SpV)<r (3) 
Imagine p also written p = H(£i v 0i + h V 0 2 + h V 0 3 ) 
D,p = D* log H.p + H (DA- V 0i + DA* V 0 2 + DA. v 0 8 ) + (Sp V )<x 
h r 
whence D,p - D t log H.p = - 4 DA* v 0i + DA- V 0 2 + DA- V 0 8 ), 
or 2D f g = DA* V 0i + &c (4) 
These formulae enable us to find convenient forms for certain 
expressions, giving the manner of the motion. Thus 
D ( Sg = SD (< r.g + S«TD 1 £ 
= S( V D,P + Vp<r + S^v <7)4 - S<7S4 V <7 by (1) 
XX xi 
= SvDjP.T (5) 
since 
S(pSo-v - <xSp v )(T = SV. Vpo- v (t = SVp<7 Vff = 0 
and generally 
S(pS() v -SSpv)<r = 0 (6) 
whatever vector 3 may be. Again 
D, ^ = S; H" S( l SffV+ffS H V)ff 
= StTg - 2S^.(S<7 V )<7. (7) 
Also writing 
i ' L E tM R* N B 
as in (2), we get 
S3 V = -(Li + M-t+N-t) 
d(pi dcpz d<f> «/ 
M^ + *& D ‘ +N 4M 
= S(D e S + Sav<7)v +SavD ( by (2) 
and especially 
Dt ' S H v =(S 3 v)Dt 
but 
D t (Sff V ) = (Str V ) + (So- V )D<. 
( 8 ) 
