206 
a da 
,( 6 ) 
, • 1 , • 
■whence we obtain v o- = v + V (t>2'jT- + V 0s^-~ 
Cv(j)i Cl^2 ^03 
n 
= y + V + V ^>aD(g 
( 7 ) 
a /3 y S( V0i« + V«i2/3 + V^av) 
Nowvi;.ijj+ Vc/>2g+ V^>8g = -^-^^ 
+ 
V(V0ia + V02/3+ V^sy) 
H 
of which the vector part is seen to vanish and the scalar 
becomes - 3. Whence 
T-xCt t^/ 3 -r. Ctx^ y 
V0lD,g+ V<?)2Dt|J+ V03D^||= -DiV02||-D^V03g 
and HVvo- = V(aD^v0i + /3DiV02 + yDiV03) (8) 
Of these (7) gives the more convenient form, from which 
we obtain 
H V V <T = V V <^iDiV V 02 V ^3 + &c. 
= V V 0 lVD^ V 02- V 03 + V V 0 lV V 02D« V 03 + &C. 
and applying to each of these expressions the formula 
VaV/3y — ySa/3 — /3Say, 
we obtain 
0gv{Sv0Av02 - Sv02r>tV0i} + &c. 
+ DeV03{SV0lV02 - Sv0iV02} + &C. 
= V0l{SV02Di03 - SV03DtV02} + &C. + &c. 
Whence S v02DfV03 = Sv03D<V025 &c. , if the motion is irrotational 
or expressing 
H V'^o’ = lici + + l^y 
we find by operating with Sv0i on each side — 
- nil = Sv0iV 02D«/3 + Sv0iV03b)iy 
= S(yD,/3~/3D,y) 
Whence — HWvo- = S (yD^/3 - /3D^y) a + <fec. + &c (9) 
and Vv'T = S 
{ 
H^'H 
and D 
,s( 
~T) ~ — ~T) ~ i— S 
ft d 
D. 
= sl ^ D' 
( H f/02 H d(j)y 
{h 
.} = 
H H ‘ H 
s.(Vv0iv)P,<T 
H 
} 
( 10 ) 
