618 
NOTES AND REFERENCES. 
t. ii. (1816), Table iv. and Traite des Fonctions ELliptiques, t. n. (1826), Table iv. 
at intervals of 30' from 0° to 90°, to twelve decimals and fifth differences. But 
the march of the function is somewhat disguised by the argument being taken in 
degrees and minutes and the function in abstract number. I have in the paper “ On 
the orthomorphosis of the circle into the parabola,” Quart. Math. Jour. vol. xx. (1885), 
pp. 213—220, see p. 220, given the table (at intervals of 1° to seven decimals) 
exhibiting the argument and the function each of them in degrees and minutes and 
also in abstract number. 
335. Besides the 13 numbers mentioned by Gauss it appears by the paper, Perott, 
“ Sur la formation des determinants irreguliers,” Grelle, t. xcv. (1883), pp. 232—236, that 
in the first thousand the determinants — 468 and — 931 are irregular. 
341. Consider the equation of a curve as given in the form y — f(x) = 0; then in 
the notation of Reciprocants (t = y', a = \y", b = ^y"\ c = y"", d — ^y'"", where the 
accents denote differentiation in regard to x) the equation of the conic of five-pointic 
contact at the point (x, y) of the curve is 
a 4 (X — x) 2 
+ a 2 b (X — x) {Y — y — t (X — x)) 
+ (ac — b 2 ) {Y — y — t (X — x)) 2 
— a 3 [Y — y — t (X — #)} = 0, 
which I verify as follows: writing X = x + 6, we have 
Y = y + td + aO 2 + bd 3 + cd 4 + dd 5 , 
and thence 
Y — y — t{X — x)= ad 2 + bd 3 + cd 4 + dd 5 . 
Substituting these values and developing as far as d 5 we find 
a 4 d 2 
+ a 2 bd (ad 2 + bd 3 + cd 4 ) 
+ (ac — b 2 ) (a 2 d 4 + 2abd 5 ) 
— a 3 (ad 2 + bd 3 -f cd 4 + dd 5 ) = 0, 
viz. this is 
O0 2 + 0# 3 + O0 4 — a (a 2 d — 3abc + 2b 3 ) d 5 = 0. 
The equation is thus satisfied as far as d 4 , showing that the conic is a conic of 
5-pointic contact; and it will be satisfied as far as d 5 if only a (cdd — 3abc + 2b 3 ) = 0. 
The value a = 0 belongs to an inflexion, and reduces the equation of the conic to 
[Y — y — t(X — x)} 2 = 0, viz. this is the stationary tangent taken twice, which is in an 
improper sense a conic of six-pointic contact: the other factor determines a sextactic 
point, viz. we have a 2 d — 3abc + 2b 3 = 0 as the condition of a sextactic point. 
We might from this form, which belongs to the curve as given by the equation 
y—f(x) — 0, pass to the form belonging to the curve as given by the equation