192 
ON SKEW SURFACES, OTHERWISE SCROLLS. 
[339 
Lemma employed in the following Annexes 2 and 3. Formula} for the order and 
iveight of certain systems of equations. 
Let a a ' denote a function of the degree a in the order variables (x, y,..), and 
of the degree a! in the weight variables {x, y',..), and so in other cases; and con 
sider first the equation 
®a' > (& A^ a ' AA ’ > ••• 
/3/3', (/3 + A)fi’ +A > , 
= 0, 
where the matrix is a square; then 
Order = 2a + %A, 
Weight = 2a' + 2 A'. 
Consider next the system 
a a', (« + A) a ' +A ', (a + B\ +B ,... 
/3/3', {ft + A)f +A ’, {ft + B) P ' +b , 
= 0, 
where the matrix is a square +1, that is, the number of columns exceeds by 1 the 
number of lines; then 
Order = 2 AB — 2a/3 + 2a (2ri 4- 2a), 
Weight = (2 A + 2a) (2 A' + 2a') -2 AA' + 2aa'. 
And again, the system 
a tt ', (a + A) a ’ +A ’, (a + -BV+b', (a + C)a. r +c'> ••• 
ftp, {ft + A)p’ +A ', {ft + B)p +B , {ft + 0)p + c, 
where the matrix is a square + 2, that is, the number of columns exceeds by 2 the 
number of lines; then 
Order = 2ABG + 2a/3 7 + 2a (2AB - 2aft) + ((2a) 2 - 2aft) (2A + 2a), 
Weight = [tAB - 2a/3 + 2a (2A + 2a)} (2A' + 2a') - (2A + 2a) (2AA / -2aa / )+2A 2 A , +2a 2 a / .