40 
ME. W. SPOTTISWOODE ON AN EXTENDED FOEM OF 
Let there be any number of variables iPj, .. , and let 
and Di, D,-, .. corresponding expressions when the ^5 •• operate only on some extra- 
neous subject of differentiation, and not on x^, x^, .. so far as they appear explicitly in 
the values of D^, Dy, . . 
Then it will be found that 
DyD,=V,V.-V,. 
Dj, Dy Dj = Vy ViVi — Vij Vi ~ VjV ki ~ ViV/i + V kji + V ju y 
and so on for any number of Ds and Vs. There are special cases in which these ex- 
pressions take a more symmetrical form. Thus, if the second condition of the system 
p(j,k)=p(k,j), p(kj)=p(i,k), p(i,j)=p(j,i) 
be satisfied, the group Vkjy Vk? Vji may be replaced by the group Vkj, Voc, Vy,-, in which 
the suffixes are cyclically arranged. If the first and last are satisfied, the group may be 
replaced by Vit, Vm 5 Vty If, besides, the first condition of the system 
be satisfied, the expression for DiDyD, may be written 
VfcV/Vi — ViVii — Vj\/ik — ViVji+2V;^'i5 
or 
VtVyVi — ViVjk — V^Vw — VitVtj'“l“2Vi^s5 
according as the second, or the first and last conditions of the first system are satisfied. 
And similarly, if the two last conditions of the second system are satisfied, the expression 
may be written 
V fc V ;V i — V ijV i — V iifc Vj — V /t V H~ 2 V;'i it 
ViVjVi V_;-jfcVi VittVy- ^i/7k~\~'^^Jik’ 
If the accents in the symbols Vi, Vj, • • are understood to imply that the suffixes i,J, . . 
and not the Vs are to be combined, e. g. 
(a, b, c, dXVMT 
= «Vp -\-b{ Vj2i + S/jij + Vi;-2) + c ( Vji2 + V V p) + d Vi3 , 
