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493 
36 ON EVOLUTES AND PARALLEL CURVES. [493 
It follows as regards the relations of the given curve to the points I, J, and the 
effect thereby produced on the evolute, we only need to consider the case of a single 
branch ; viz. the cases are 
the given curve intersects the line IJ at a point other than / or J, and belonging 
thereto there is a branch ordinary or singular, 
not touching IJ, 
touching IJ; 
and the given curve passes through the point I or J, and belonging thereto there 
is a branch ordinary or singular, 
not touching IJ, 
touching IJ. 
I have succeeded in determining the effect, not for a singular branch of any kind 
whatever, but for branches of the form y = x k , y* _1 = x k ; viz. k = 2, each of these is an 
ordinary branch, k= 3, the first y=x? is an inflexional branch and the second y = x§ a 
cuspidal branch ; and so k > 3 the two branches are respectively inflexional and cuspidal 
of a higher order. I do this very simply by consideration of the curve cJ~ Y z = y k . 
The curve in question x k ~ 1 z = y k , is a unicursal curve, and it has a reciprocal of 
the same form X k ~ 1 Z= Y k , hence 
m = n = k ; 0 = ft — 2m +2 + k, 
whence 
t — k = k — 2, t = 8 = ^ (k — 2) (k — 3); 
viz. the point x = 0, y = 0 is a cusp equivalent to k— 2 cusps and ^(k— 2){k — 3) 
nodes; and the point z = 0, y— 0 is an inflexion equivalent to k — 2 inflexions and 
\ (k — 2) (k — 3) bitangents. 
The equation U = x k ~ l z — y k = 0 of the curve is satisfied by writing therein 
x : y : z = 1 : 6 : 6 k ; and these values give 
d x TJ : d y U : d z U = (k — 1) z : —ky k ~ l : x k ~ x , =(k—l)6 k : —kO k_1 : 1. 
Taking the coordinates of I, J to be (a, /3, 7) and (a', /3', y) respectively, and 
X, Y, Z as current coordinates, the equation of the normal at the point (x, y, z) of 
the curve 17 = 0 is readily found to be 
(a'd x U + {3'd y U + v'd z U) 
X, 
Y, 
z 
a , 
/3, 
7 
x , 
y » 
z 
+ d x U + /3 dyU + 7 d z U) 
X, 
Y, 
Z 
a , 
¡3', 
r 
7 
x , 
y > 
2 
or, € 
wher 
the 
none 
term 
consi 
equal 
the 1 
with 
= 3 k 
throv 
= m - 
equal 
writii 
co, ir 
= co - 
out; 
with 
a un 
in th