14 
ON ARONHOLD S INTEGRATION-FORMULA. 
[632 
which, substituting therein for P, Q, R, P 0 , Q 0 , R 0 , their values, and writing 
(A, /a, v) = (r)-y, x-%, %y-yx), 
is in fact 
ATF= (A, ...$A, fi, v^a, /3, y). 
We have identically 
(a, ...$#, y, 1 ) 2 . (a, ...$£, y, l) 2 - W 2 = (A, ...£A, ¡x, v) 2 , 
which, in virtue of (a, y, 1) 2 = 0, gives 
W 2 =-(A, ...$A, /x, i/) 2 ; 
and since A 2 = - {A, ...]£a, /3, y) 2 , the equation is thus 
V{-(4. •••$«, & 7) 2 }-•••$>. AL v) 2 } ={A, ...}[A, /a, v$a, ¡3, y), 
(A, /3, y) 2 . (A, ...$A, y,, v) 2 -[(A,...Q\ fi, i/£a, /3, 7)] 2 = 0. 
The left-hand side is here identically 
= K (a,.. .][y/x — /3v, av — yX, /3A — a/A) 2 : 
substituting for A, /a, v their values, we find 
(y/x — fiv, av — y\, ¡3\ — ol/jl) = (a;fl 0 — £il, yil 0 — yil, z£l 0 — §12); 
viz. in vii’tue of il 0 = 0, these are = — fO, — 77il, — £0, and the quadric function is 
= ifil 2 (a, ...$£, y, l) 2 , vanishing in virtue of the relation (a, ...$£, y, 1) 2 = 0. 
The equation in question 
V{—(-4•••$«» & y) 2 } .<\/{—(A...][A, /A, v) 2 } =(A...][A, /a, v'fra, /3, 7) 
is thus verified, and the theorem is proved. 
633] 
NOTE