[895 
or a 4 , we obtain in each 
of 24, viz. 6, 4, 1 in the 
as, viz. there is 1 tree 
thus proved. Hence the 
to see that the proof is 
896] 
29 
given knots A, a, /3, y, 
nots A,, /x, a, /3, 7, ..., the 
a, /3, 7,...) the triads of 
A, fi, v always belonging 
of the knots A, p, v, ..., 
trees or pairs, or triads, 
then n being the number 
of knots A, a, ¡3, 7, ..., 
here i= 1, n — 2, and we 
3S are 
(A/3, pa). 
= 4 x 1.4 2 = 64 
+ 6x2.4' 48 
+ 4x3.4° 12 
+ 1 1 
125, 
branch a/3 to any tree 
+ 7, 8, e); the branches 
stly the tree a(3. ay. a8. ae: 
.ble manner. The whole 
; the theorem is of course 
oer by Borchardt, “ Ueber 
lations-Resultante,” Crelle, 
¡rs a certain determinant, 
.., nO, n\, ..., nn— 1, and 
anches of the tree being 
al to the product of the 
is thus =(n+l) }l_1 
896. 
A TRANSFORMATION IN ELLIPTIC FUNCTIONS. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xxiv. (1890), 
pp. 259—262.] 
The formula in question is given in Klein’s Memoir “Ueber hyperelliptische 
Sigmafunctionen,” Math. Ann. t. xxvii. (1886), pp. 431—464, see p. 454, in the form 
u 
- r co (1A - Vf f(y)\+F(x, y) 
"kvîTW e( ) ~ 2(*yr ' ■ 
(the discovery of it being ascribed to Weierstrass) ; and it is also given in Halphen’s 
Traité des Fonctions Elliptiques, t. 11. (1888), p. 357. 
The algebraic foundation of the theorem is as follows : writing 
z = \ [f{ae) + c}, 
and as usual I, J for the two invariants of the quartic function (a, b, c, d, efx, y) 4 , we 
have identically, as is easily verified, 
4z z — Iz — J = [d f(a) + b \f(e)Y, 
or say 
Hence putting 
V(40 3 — Iz — J)= d f(a) + b \J(e). 
A = (a, b, c, d, e\x 1} îc 2 ) 4 , 
B=(a, ##1, xtf(y it y 2 ), 
E=(a, ^y u y 2 ) 4 ; 
A = Xi y2 x 2 yi, 
as 
and