158 
PROFESSOR M. J. M. HILL ON THE LOCHS OF SINGULAR POINTS 
Put, therefore, 
f = R {a — eq) (a — a 2 ) (a — ot 3 ).(64), 
where cq, a 3 , a 3 become equal when x — £ y = 17 , 2 = £. 
Therefore 
P/ 
Da 
DR / \ / x / \ 
— (a - (a - a r ) (a - a 3 ) 
+ H [(« — a i) (a — a 3 ) + ( a “ * 3 ) i a — a i) + (a — «i) (a — a 2 )} . (65), 
L 2 
Da 2 
D 2 R 
Da 2 
(a — 
a i)(« ~ a a) ( a ~ a s) 
DTI 
+ 2 ((« “ a i) ( a — «s) + (a — a 3 ) (a — cq) + (a — a L ) (a — a 3 )} 
“h 2 Pt {(a — eq) -f- (a — « 3 ) -p (a — a 3 )}.( 66 ). 
Now, at a point on the locus of binodal lines, the two equal values of a which make 
1 )//Da = 0 , become equal to the same thing as cq, a 2 , a 3 . 
Hence /(x, y, z, a 2 ) = (R') (a 3 — eq) (a 3 — a 3 ) (a 3 — a 3 ), where R' is what R becomes, 
when a is changed into a 0 . 
Hence f(x, y, z, a 2 ) is of the third order of small quantities ; but D~f(x, y, z. a 1 )/Da 1 2 
is of the first order, for the most Important term in it is 
Hence 
2 (R) [(oq cq) -|- (cq — eq) -f- (cq — a 3 )]. 
/(*, y> 
n . \ /D 2 / (x, y, z, a x ) 
a ‘>! iv “ 
( 67 ) 
at points on the locus of binodal lines. 
In like manner 
but 
f[x, y, z , a 
• VF ^}‘= 0 
f(x, y, z, a 2 ) 
/ [ D 2 /( x - y, «i ) 
/ 1 IV 
3 
^0 
( 68 ), 
( 69 ) 
at points on the locus of binodal lines. 
