454 
PEOFESSOE CAYLEY ON THE TE AN SFOEM ATI ON 
III. 
y 6 
y 5 
y 4 
y 3 
y 2 
y 
1 
ar 6 
+ 1 
x s 
— 256 
+ 320 
- 70 
X 4 
-640 
+ 655 

a 3 
+ 320 
-660 
+ 320 
+ 655 
-640 
X 
- 70 
+ 320 
-256 
1 
+ i 
+ i 
- 6 
+ 15 
- 20 
+ 15 
- 6 
+ 1 
= 0 - 1 ) 6 - 
IV. 
y ' 6 
y 5 
y 4 
y z 
y 2 
y 
1 
X 6 
+ 1 
X* 
- 65536 
+ 163840 
-138240 
+ 43520 
- 3590 
x i 
+ 163840 
-133120 
— 207360 
+ 133135 
+ 43520 
X 3 
— 138240 
—207360 
+ 691180 
-207360 
— 138240 
+ 43520 
+ 133135 
-207360 
— 133120 
+ 163840 
X 
- 3590 
+ 43520 
-138240 
+ 163840 
— 65536 
1 
+1 
+ 1 
- 6 
+ 15 
— 20 
+ 15 
- 6 
+ 1 
=(y-l) 6 . 
where the subscript line, showing in each case what the equation becomes on writing 
therein x= 1, serves as a verification of the numerical values. 
78. The curve I. has at the origin a dp in the nature of a fleflecnode, viz. the two 
branches are given by # s + 4y=0, — ^ 5 +4^=0 respectively; and there are two singular 
points at infinity on the two axes respectively, viz. the infinite branches are given by 
—y— 4.r 5 = 0, x— 4?/ 5 =0 respectively. Writing the first of these in the form 
— yz 4 — 4# 5 =0, we see that the point at infinity on the axis #=0 {i. e. the point z= 0, 
#=0) is =6 dps; and similarly writing for the other branch xz*— 4z/ 5 =0, the point at 
infinity on the axis y = 0 ( i . e. the point 2 = 0 , ^=0) is =6 dps*. 
Moreover, as remarked to me by Professor H. J. S. Smith, the curve has 8 other dps ; 
* These results follow from the general formulae in the paper “ On the Higher Singularities of Plane Curves,” 
C. & D. M. J. t. vii. (1865) pp. 212-222 ; but they are at once seen to he true from the consideration that the 
curve yz i —x i = 0, which has only the singularity in question, is unicursal ; the singularity is thus =6 dps. 
