364 ME. W. SPOTTISW OODE ON HYPEEJACOBIAN SUEPACE3 AND CTTEVES. 
Operating a second time : 
(t : m) A 2 P + 2D'AP : m+2(AD')P : m+2D'AP : m + 4(A, . . • • •) : m 
=A 2 P 0 +2mj2+p 1 : (tz-1)}{P 1 AH+HAP 1 +2(A, . . .)PA • • • ••)} 
+m{4HAP 1 +UA 2 P 1 +2 i +HAP 1 :(w-l)(. 
But, in virtue of (3) and (6), 
(AD)P=0, (A, . . OPJE, . . OPJPi, • • 0=0, (A, . , opjy, • • -XW • • 0=0 ; • (8) 
and consequently 
(tf:m)A 2 P + 4mHAP 1 :(w-l)=A 2 P 0 +2mj4+Q? 1 +j?/):(w-l)}HAP 1 . . (9) 
Again (writing down only the first lines of the determinants), since 
AP 0 = -HPi+U, A (w, . . .), (10) 
A 2 P 0 =-HAP 1 -P 1 AH-2(A, . . ,)(d x H, . . .)(dj\, . . .) 
+ 4H, A (w, . . .) +U, A 2 (w, . . .)+2 w, A'A(w, . . .) 
+2v, B'A(w, . . 0 
+2w, C'A(w,...) 
+ 2£, D'A(w,...); 
or omitting terms which vanish, 
A 2 P 0 = — HAP,+4H, A(m, . . — 8H, A(«, . . ,)+2A'{tt, A(«, . ..)( 
+2B>, A(«,...)} 
+ 2C'{i(I; A(a, ...))■ 
+2 D'{£,A(m,...)}- 
But 
w, A(w, . . .)=A(u, w , . . .)— Aw, w, . . . — 2(A, . . .)(«» «/, fl')(d« • • 0( M > • • 0 » 
and of these terms A (w, w, . . .) vanishes identically, and A u, w, . . . contains the three 
columns u,v,w; so that A'{ A w, u, . . .] will contain two of them, and will consequently 
vanish. 
But if we retain only terms which contain not more than two of the columns w, v , w, Jc, 
and which, after the operations A', B', C', D', will consequently contain only one such 
column, we shall have 
w, A (w, . . .)=2 w, A 'u, w w, . . .+2w, Aw, v, v', . . . 
+2w, B 'u, v x , w, . . . + 2w, B 'w, v, u ', . . . 
+ ... +... 
and consequently 
A '{u, A (w, . . .)}=2w, Aw, w', A w, . . .+2w, Aw, Afl, f/, . . . 
+2w, B'w, Vi , Aw, . . . + 2w, B'w, A'u, w', . . . 
+... +... 
