205 
The condition of continuity is therefore that — 0 
where p denotes the density, and this condition in an incom- 
pressible fluid becomes DiH=0. Generally where 
D(0=O. 
Before proceeding to investigate any forms for o-, the 
velocity, we will find some expressions from the action of 
Dt on the expressions just found. 
d 
Remembering that ~ ^ V we see that 
D^v = vB<+ a(So-v) 
symbolically, where A is the same operator as v but acts 
on the or only. 
Thus D^V0= A(SffV0) 
f^.d(T n,d(T rjdcf 
— V + V V ^ + V V 0 4- V V 0 .... (4) 
since V = V + V 0.^^ + V 03^^ 
[These other two forms of v will also be used 
— H V = V 0lS V 02 V 03 V + V 02^ V 03 V 01 V + V 0sS V 01 V 02 V > 
or =aSv0iV +/3 Sv 02V +ySv03V, 
since v is in form a vector.] 
Let us first apply these results to S v 0i V 02 V 03 or - H 
- DtE = SDe V 01. V 02 V 03 + S V 0lDj V 02. V 03 + S V 01 V 02 V 03 
'da da da 
\d(f)i 
= - Hsf 
or iD,H 
H ^ 
) 
Sv«r= - HD 
a) 
,(5) 
or is evidently rate of compression per unit of volume. 
If again we act similarly on a or V V 02 V 03 
Dftt = VD^ V 02. V 03 + V V 02Be V 03 
- S V 02^ V V 01 V 03 + V 02^ V V 02 V 03 
+ V 03^ V V 02 V 01 + V 03^ V V 02 V 03 
= SV<r.a-S^Vi.i.a-S^V<p,.(}- V ?>s.y 
= S Vff.a + Htt 
d(pi 
. 1 T^ T^ ^ 
• • H^‘“ ^ - df, 
