NOTES AND REFERENCES.
587
it would have been proper to notice the distinction between actual and apparent
double points (first made by Dr Salmon, to whom the term apparent double point,
adp, is due) : and the like in regard to the lines in two planes. In the translation of
this paper, 83, there are two footnotes signed G. S. (Dr Salmon), giving for plane curves
the formulae l — k = 3 (v — g) and 2 (t — S) = (v — g) {v + /a — 9) : and for curves of double
curvature and developable surfaces the analogous formulae a — ¡3 = 2 (n — to), x — y = (n — to)
and 2 (g — h) — (n — m)(n + to — 7). Also in the second set of six equations, p. 210, the
last three equations are replaced by
x = |to (to — 1) (to 2 — to — 4) — ^ (2A + 3/3) (2h + 3/3 +1) — to (to — 1) (2h + 3/3) + 3h + 4/3,
a = 2to (3to — 7) — 3 (4 h + 5/3),
g = \m (3to — 7) (3to 2 — 5to — 7) + \ (6A + 8/S) (6A + 8/3 +1) — 3to (to — 2) (6A + 8/3) + 19A 4- 24/3,
viz. these are the equations serving to express x, a, g in terms of to, h, /3.
For the discussion of some singularities not considered in the present paper see
Zeuthen, “ Sur les singularités ordinaires des courbes géométriques à double courbure,”
Comptes Rendus, t. lxvii. (1868), pp. 225—233, and “ Sur les singularités ordinaires
d’une courbe gauche et d’une surface développable,” Annali di Matem. t. ni. (1869-70),
pp. 175—217.
40. See 10.
41 and 44. For demonstrations of Sir W. Thomson’s theorem for the value of the
f dec. • •
definite integral /77 rr ‘V. ,■ „ ttttt see his papers “ Demonstration d’un
5 J {(x-a) 2 +... + u 2 } l (x>+...+v 2 ) l+1
theoreme d’Analyse ” and “ Extrait d’une lettre a M. Liouville,” Liouv. t. x. (1845),
pp. 137—147 and 364—367, also the paper “ On certain definite integrals suggested by
problems in the theory of Electricity,” Camb. Math. Jour. t. 11. (1845), pp. 109—121.
45. I am not aware that the equation Jk sn u = H (u) a © (u) had been previously
demonstrated otherwise than by the circuitous process employed in the Fundamenta
Nova.
The series z
rp2 rrd'
1 + O + c, T7¥7Tr
+..., which is the solution of the differen-
tial equation x?z + ax ^ ^ — 2 (a 2 — 4)^ = 0, and in which C 1} C 2 , ... denote the coeffi
cients of the highest powers of n in the expressions given p. 299, is in fact (as
remarked by me in a later paper, Liouv. t. vii. (1862)) the Weierstrassian function
A1 (x).
47. The surface here considered, the Tetrahedroid, is the general homographie
transformation of the wave surface. It is a special case of the 16-nodal quartic surface
considered by Kummer in his Memoir, “Algebraische Strahlen-systeme,” Berl. Abh. 1866,
pp. 1—120, and in various papers in the Berliner Monatsberichte.
74—2
