538 
PEOFESSOE CAYLEY’S EIGHTH MEMOIE ON QUANTICS. 
tions give J= — , (x, y) within the region P ; the corresponding character being 5 r, 
which is right. 
2 °. y— -j- , \|/(#, y)=—, the point (x, y) must be within the loop, or within the triangle ; 
if ( x , y) is within the loop, theny — 1 = + ? ®-Vy— 1 ? and the condition J(y—l)(x -\-y)= + 
becomes J= — , that is, we have J= — and ( x , y) within the loop, that is, in the region T. 
And again, if (x, y) be within the triangle, then y — 1 = — , x-\-y =-\ -, and the condition 
J (y — 1 )(x-\-y)=-\- still gives J= — ; but J= — , and (x, y) within the triangle, that is, 
in the region T or P, will of necessity be in the region T ; so that in either case we have 
J=— , (x, y) in the region T, which agrees with the character r-\-U. 
3°. y=+, ip(x, y)=-\~, ( x , y) is in the upper region, that is, in the region Q or T ; 
if {x, y) is in the region Q, then of necessity J= — , and if in the region T, then of neces- 
sity J= -j- , that is, we have 
J = — . (x, y) in the region Q, or 
J = + , (x, y) in the region P, 
which agrees with the character r+ 4 i. 
And it is to be observed that the portions of T under 2° and 3° respectively make up 
the whole of the region T, and that 3° relates to the whole of the region Q, so that the 
conditions allow the point ( x , y) to be anywhere in Q or T, which is right. 
4°. y =-\ -, -ty(x, y)= — , ( x , y) is in the loop or the triangle, and then y — l=-f 
implies that it is in the loop, whence x-\ -y=-\~, and the conditionJ [x-\-y )= — becomes 
J= — ; we should therefore if the combination existed have J= — , (x, y) within the loop, 
that is, in the region T ; but this is impossible. 
303. Hermite’s second set of criteria are 
y= +, y=+, J=— , ^(x,y)=—, character 5r. 
y=+, V— y=+> J =-> 'Pfay )=+ 1 
y =+ , \ r— y=+, J=+ i character r+4f. 
y =+ , V-y=-. I 
1°. If x , y) — — , then the point (x, y) must be situate within the loop or 
within the triangle ; and recollecting that at the highest point of the loop we have 3 /=^, 
the condition — y= + is satisfied for every such point, and may therefore be omitted. 
The conditions therefore are J= — , (x, y) within the loop, that is, in the region T, or 
within the triangle, that is, in the region P or the region T ; but for any point of T the 
general theory gives J= +, and the conditions are therefore J= — , (x, y) within the 
region P ; which agrees with the character 5 r. 
2 °. y~-\-, ^{x, y) — -\-, that is, (x, y) is within the upper region, that is, in the region 
Q or T ; and y— + , ( x , y) will be within the portions of Q and T which lie beneath 
the line y=- 9 -; but J=— , and therefore (x, y) cannot lie in the region T ; hence the 
conditions amount to J= — , (x, y) within that portion which lies beneath the line y=-<r 
of the region Q. 
