68 
ALGEBRAIC INVARIANTS 
Hence 
VafP 2 = 4a$/3,,(a/3), V 2 a^P 2 = 4 (a/3) 2 , 
VatPta r ,P r , = Pta ri - VatPq+OiiPr,-VP k a v 
= P^r,(a0) -\-a$ v (Pa), 
V 2 a i P i a n P v = (/3a) (a/3) + (a/3) (/3a) = — 2 (a/3) 2 . 
The difference of the expressions involving F 2 is 6 (a/3) 2 . Hence 
if (1) operates twice on the equation preceding it, the result is 
6(a/3) 2 = 12K, K = i{aP) 2 . 
43. Lemma. F n (^) n = (w+l)(w!) 2 . 
We have proved this for n = 1 and w = 2. If w > 2, 
07?2 
( {•))■ = «( !•))"+»(»■-1) ({<!)•■"V fl. 
okiovz 
Similarly, or by interchanging subscripts 1 and 2, we get 
0120771 
Subtracting, we get 
F(^) n = l2»+w(»-l)}(^T?) il - 1 =«(«+l)(^) n - 1 . 
It follows by induction that, if r is a positive integer, 
V r {£v) n = {n+l)[n{n — 1) . . . (n — r-\-2)\ 2 {n—r+l)(^i?) n-r . 
The case r = w yields the Lemma. 
44. Lemma. // //?£ operator V is applied r times to a product 
of k factors of the type and l factors of the type /3,, there results 
a sum of terms each containing k—r factors a$, l—r factors /3,, 
and r factors (a/3). 
• The Lemma is a generalization of (2), § 42. To prove 
it, set 
A =af 1 ' > af 2 ^ . . . af k \ 
B=t3, (1 W 2) ■ ■ • o,®.