[822 
823] 107 
e thus have 
823. 
ON THE GEOMETRICAL INTERPRETATION OF CERTAIN 
FORMULAE IN ELLIPTIC FUNCTIONS. 
uble algebras 
nt, A merican 
lr C. S. Peirce 
ixture of two 
lucible to the 
in the simple 
between the 
he system of 
multiplication 
[From the Johns Hopkins University Circulars, No. 17 (1882), p. 238.] 
I HAVE given in my Elliptic Functions expressions for the sn 3 of u + \K, u + \%K', 
u + \K + \iK'; but it is better to consider the dn 2 , sn 2 , cn 2 of these combinations 
respectively, and to write the formulae thus : 
, i 7n _ ij dn u ~ G - k> ) 811 u cn u _ y 1 - “ (1 ~ k ')y . 
2 ‘ dnM+(l -A')snwcnM ’ ' 1 — k 2 x + (1 — k') y ’ 
/,, , i „• v>\ 1 (1 + k) sn u + i cn u dn u 1 (1 + k) x + iy 
( u + 2 1 j c (1 + k) sn u — i cn u dn u ’ k (1 + k) x — iy 
a , , Tr . . jr/\ — ik' cn u — (k + ik') sn u dn u — ik! 1—x — (k + ik') y 
cn 2 (u+ kK + —. 77 t , =—7—3 U-—; 
k cn u + (k + %k ) sn u dn u k 1 — x + (k + ik ) y 
where in the last set of values x, y are used to denote sn 2 u and sn u cn u dn u 
respectively; and the formulae are thus brought into connexion with the cubic curve 
y 2 = x (1 — x) (1 — k 2 x). The curve has an inflexion at infinity on the line « = 0; and 
the three tangents from the inflexion are x = 0, x = l, x = y,,, touching the curve at the 
points x, y — (0, 0), (1, 0), , 0^ respectively : hence these points are sextactic points. 
We may from any one of them, for instance the point (0, 0), draw four tangents to 
the curve, (1 + k) x + iy = 0, (l + k) x — iy = 0; (1 — k) x + iy — 0, (1 — k) x — iy = 0; where 
the first and second of these lines form a pair, and the third and fourth of them 
form a pair, viz. the two tangents of a pair touch in points such that the line joining 
them passes through the point of inflexion: in particular, for the first-mentioned pair, 
the equation of the line joining the points of contact is l+fcc = 0. The linear functions 
belonging to a pair of tangents are precisely those which present themselves in the 
formulae; thus if = (1 -1-k) x + iy, T 2 — (1 + k)x — iy, the second of the three formulae 
is sn 2 (ii + ^A) = v r .]; and the other two formulae correspond in like manner to pairs of 
r£ l 2 
tangents from the sextactic points ^, Oj , and (1, 0) respectively. The formulae are 
connected with the fundamental equations expressing the functions sn, cn, dn as quotients 
of theta functions. 
14—2