38 
ON EVOLUTES AND PARALLEL CURVES. 
[493 
The results are respectively as follows: 
m — a — 
n" = m + n — 
t" = 0 + 
— 3m - 3n + 3a — 
Ak 
Bk 
c k 
Bk 
P k 
Qk 
Bk 
Sk 
0 
3k-3 
k- 2 
k + 1 
k 
3k-2 
2k-2 
2k 
0 
k - 1 
0 
1 
1 
k 
k - 1 
k 
0 
k - 1 
k- 2 
k- 1 
k-2 
k-2 
0 
0 
0 
5k-5 
2k - 4 
2k + 1 
2k - 1 
5k - 4 
3k-3 
3k 
m = a — 
n" — m+n — 
t" = 0 + 
3m — 3n + 3a — 
a 2 c 2 
B 2 D. 2 
A 3 
B 3 
o 3 
B 3 
r 2 r 2 
Q*s 2 
P 3 
Q 3 
B 3 
s 3 
0 
3 
0 
6 
0 
4 
3 
4 
3 
7 
4 
6 
0 
1 
0 
2 
0 
1 
1 
2 
1 
3 
2 
3 
0 
1 
0 
2 
0 
2 
1 
0 
1 
1 
0 
0 
0 
5 
0 
10 
0 
7 
5 
6 
5 
11 
6 
9 
read for instance in B k , m" = a — (3k — 3), n” = m + n — (k — 1), i" = 0 + (k— 1), and 
re" = — 3m — 3n + 3a + (ok — 5); and so in other cases. 
A. 2 G 2 (that is indifferently A 2 or C 2 ) is when there is on IJ an ordinary point, 
IJ not a tangent; and so B 2 D 2 when there is on IJ an ordinary point, IJ a tangent. 
Similarly P. 2 R 2 when there is at J an ordinary point, IJ not a tangent; only instead 
thereof I have written P 2 R 2 to indicate that (for a reason which will appear) the 
numbers are not deducible from those for P 2 or R. 2 by writing therein k = 2; and 
Q 2 S 2 is when there is at J an ordinary point, IJ a tangent. 
Case A k . We have to take the line IJ passing through the inflexion; the con 
dition for this is /3y' — /3'y = 0 : there is no speciality, or we have n" — 2&, m" = 4& — 2, 
l" — 0; whence also k" = 0; the value of re" being in every case deduced from those 
of m", n", l" by the formula 
Sm" + l" = Sn" + k". 
Case B k . I write y = y = 0, the equation of the normal is 
№ .(k—1) {(aff + «73)X- 2aa Y} 
+ O 2 *- 1 . — k {2¡3/3'X — (a/3' + a ft) Y} 
+ 6 k+1 .(k—1) 2oiol'Z 
+ 6 k .-(2k-l)(a(3'+a'l3)Z 
Yd*- 1 , k. 2/3/3'Z =0, 
e k+i (x, Y) 
+ 6 k (X, Y) 
+ 6 2 z 
Ye z 
+ z = 0, 
flj