THE INDEX SYMBOL IN THE CALCULUS OF OPEKATIONS. 
17 
= (VV.I«2/32)(VV.I«./3,XVV.I«/3)-(VV.I«./3.)(VV.Vl«,/3J«/^j 
- 2a,( VViI^./3.)( VV,I«/3) + 2«,( VV. 
- /32(VV.I/3.«,)(VV.I«^)+ /3.(V.VV,I«,/3 J43) 
- /3,(VViI«./3J(VV,Ii3«)+ /3,(V.VV.I«./3J«/3) 
=(VVI«2/32)(VV.I«,/3,)(VV,I«/3) 
-(VV.Ia (3) (VV,VX^AX^2f32) 
~(VViI«i/3.XVV.Vl«.M« i3) 
-(VV,I«2/3.XVV.VI« ^ 
+2(VV.VViI«2/3.Ic4:/3J«/3). 
The expanded form of this is 
KV+i3,Vi)(a.V+/3.V,X«V+/3Vi) 
-(a V+/3 Vj{(«.«.+/3A)V+(«A+«2/3JV.} 
-(a,V+AV0{(«2« +i3.i3 )V+K/3 +«^2)V.} 
-(a2V+/3,V.){(« «.+/3 A)V+(« /3,+a,^) VJ 
+ 2 { (aajaa +a/3i/32 + 04 ,^ 2 /^ + 042/3/3,) V 
+ (/3/3,/32+/3 c4ia2+/3, 04204 +/32ao4,)Vi } 
= aV' + 35 V Vi + 3cV V ? + dVi 
— 3 (« + c) V" — 3 ( 35 -h 6?) V Vi - 5c V? 
+2(a+3c)V+2(35+t?)Vi. 
To these may be added, 
{abcdeXE 
= (VV.I«3/33XVV.I«2/3.)(VV.Ia.^,)(VV,I«/3) 
-(VVxI«3/33)(VV /3 XVViVI«2/32X«,/3,) 
-(VV J«3/33)(VV,Io4,/3,)(VViVI« /3 X^A) 
-(VVaI«3/33)(VVjo42/32)(VV.Vl«,/3.Io4i3) 
-(VV.Io42/32)(VV:Xa:/3.)(VViVl«3/33l« ( 3 ) 
-(VViX“ }(VV.X“2/32)(VV.VX«3/33X«./^.) 
-(VViX«i^i)(VV,X“ (3 XVV.VX«3^3X«2/32) 
+2(VViX“ f3 )(VV.VV,X«3/33X«2^2X“ii3.) 
+2(VV.X^.^i)(VV,VV,X« i3 X^s(3sX^A) 
+2(VV.X“2/32)(VV.VV,X“ii3iX“ i3 X“3/33) 
+2(VV J 043 / 33 ) ( VV, VV,X«2/32X“i/3iXc‘ 
+ (VViVX« 3/33X“ i3 )(VViVX“2/32X«./3.) 
