1919-20.] On a Class of Graduation Formulae. 
119 
Considering the graduated value of u 0 as the quotient of the two 
determinants (A), it is obvious * that in the general case the curve of 
coefficients will have as its equation 
y y y y 
+) “ J 2 ^4 • • • ^2 k 
y = 
y y y 
+L ^6 • • * ^2^+2 
- 
y y y y 
^2 ^6 ^8 • • • Zj 2k+2 
2 2 2 4 2 6 ... 2 2fc+2 
y y y 
+5 ^8 * • • ^2&+4 
2 4 2 8 2 10 * • * 2 2*+4 
y y y 
^2k+2 +jfc+4 • • • ^4& 
^2 fc ^2&+4 ^2 &+6 ‘ * * ^4 
^2fc+2 -^2*+4 • * • ^4fc 
2 2 
2 2 
^4 G 
"10 
'10 
^2 fc ^ 2*+2 ^ 2*+6 ^ 2 fc +8 * ' * ^ 4 * 
* ^2fc+2 
• ^ 2^+4 
+(-i y 
and will be of this form : — 
2 2 2 4 
^ 2^+2 * 
J 2k 
' 2^+2 
■'4fc~2 
,2k 
Substituting the values 
equation (B), we have 
(a) when k— 1, 
(B), 
1 and 2 for k in 
2, - 2„# 2 , 
or 
(2 m - l)(2m + l)(2m + 3)y = 3(3m 2 + 3m — 1) — 15a; 2 ; 
(l 3 ) when k — 2, 
2 0 2 2 2 4 
2 2 2 4 2 g 
2, 2, 2 n 
y = 
2 4 2 6 
- 
2 2 2 6 
a? 2 + 
2 2 2 4 
00 
o 
00 
^ 2 6 
A 
or 
4(2m - 3)(2m - 1)(2??? + l)(2m + 3)(2m + 5)?/ = 15(15m 4 + 30m 3 - 35m 2 - 50m + 12) 
— 525(2m 2 + 2m - 3)a? 2 + 945a? 4 , 
which give the following equations to the parabolae for different values of m. 
(a) k= 1. 
m= 11, 805^/ = 79 - a? 2 , where a? has values ranging from - 11 to +11 
m= 12, 5175?/ = 467 -5a? 2 , ,, „ ,, -12 „ +12 
m = 13, 1305?/ = 109 - a? 2 , „ „ „ -13 
m= 14, 8091?/ = 629- 5a; 2 , „ „ „ -14 
m=15, 9889?/= 719 - 5a? 2 , „ „ „ -15 
* As is easily seen by expanding the elements in the first row of the numerator 
determinant in equation (A). 
„ +13 
„ +14 
,, +15. 
