18 
ME. W. SPOTTISWOODE ON AN EXTENDED POEM OE 
+ (VV,Vla3/33l«i/30(WiVlc4 i3 
+ (VViVlc43/33X«3i3.)(VViVXa,/3,Xa /3) 
-6(VViVViVXa3/33X«3i3.X«,/3,X«i3) 
= aV" + 4 J V Vi + 6cV V? + 4(Z W? + eVt 
-6{(«+c)V^+2(25+(Z)VVi+(5c+e)VV?+2eZvn 
+ (ll«+30c+3(Z)V"+44(^+fZ)VVi+4(6c+2(7+3e)V? 
— 6{(«4'6(?+e)V+4(^+^)Vi}- 
There is considerable resemblance between these functions and determinants ; and 
although the factors of which the various terms are composed do not admit of being 
arranged as constituents of the determinants which they resemble, yet a s)'mbolical no- 
tation will render such an arrangement possible. For this purpose, let ( VViX^./^iX 
Xa>i3,) be the symbolical factors of (VViVX“<i3«X“ift) ’ 
( VV:X«2i33l X I«.i30 = ( VViVX«3/33Xai/30 
(VViIa3i32l X I«,i3.X X X«i3) = (VV.VV.I^^3/33l«./3,Xa/3), 
and so on. Then 
{a b cX’B.'E.y 
= (VV.I<«,/3,) I»/3) 
(VV.I»./3,I (VV.I»/3) 
(a i cXaKtf 
= (vv.IccA) x-^A) I“/3) 
(VV.I^AX (VV,Ia,/3,) Xk/3) 
(vv.laAX (vv,Io!./3,I (vv,i«!^) 
By means of these formulse, the corresponding expressions for higher degrees may be 
established. Thus, for the fourth degree, 
(ahcdeja 
= (a3E-f/33Si)(a2H+/32HiXa,H+/3iEi)(aE+/3E) 
= (a3V4-/33V,)(«2H+/32E,)(aiE-l-/3,E,)(aE+/3E,) 
— 3a3(o52E-f/32E,)(a,E-f/3iEi)(aE-l-/3Ei) 
— i33(i32E+«3H.)KE+/3.E0(«E+/3E ) 
— /33(/32E-f/32E,)(/3,E-f aiE,)(aE-f/3E,) 
— /33(a2a+/32Ei)(aiE+/3jEi)(/3E+aEi) 
=(«3V+/33V,X«2H+/32E,)KE+i8,E0(«H+i8EJ 
— (EEiEXa3/33X«2/32XaiE+/3,E,)(aE+/3E,) 
-(a2H-f/32EJ(EE«X«3i33l«,/30(«E-f/3EJ 
-(«2H+/32E0KE+^,E0(EE SX«3i33X«i3) 
