434 ON THE TRIPLE ^-FUNCTIONS. [699 
or substituting for A and B their values, and disregarding the constant factor \{a — b), 
this is 
AB = {{y — z)^/a — y.b — y.a-z.b—z.c — x.d — x.e — x./— x.g — x 
y — Z , Z CG • CC y 
+ (z -x)*/a— z .b— z.a-x.b-x.c-y.d-y.e-y.f-y.g-y 
+ {x — y)*Ja — x.b — x.a — y.b — y.c— z.d— z .e — z./— z .g— z}. 
We have thus in all 21 equations, which exhibit the form of the Weierstrassian 
functions al (u, v, w) 12 , ..., al (u, v, w) 67 . 
To complete the system, there should it is clear be 35 new functions al (u, v, w) 123 , 
..., al(w, v, w)567, represented by ABC,..., EFG, viz. the whole number of functions would 
then be 
7 + rl + rTO (=7 + 21 + 35) = 63, =64_1 - 
since the functions represent ratios of the ^-functions. 
II. 
Starting now with the radical 
Va — x.b — x.c — x.d — x.e — xx.g — x. h — x 
composed of eight linear factors, and writing, as in my “Memoir on the double 
^--functions,” t. lxxxv. (1878), pp. 214—245, [665]; a, b, c, d, e, f, g, h to denote 
these factors, and similarly a l5 b l7 c l5 d l7 e 1} f 1} g 1} h x and a 2 , b 2 , c 2 , d 2 , e 2 , f 2 , g 2 , h 2 to 
denote a-y, b-y, etc., and a - z, b-z, etc., so that X = abcdefgh, F = a^c^e^h!, 
Z = a 2 b 2 c 2 d 2 e 2 f 2 g 2 h 2 ; then, instead of the Weierstrassian form, the differential equations 
may be taken to be 
7 dx dy 
d “ = VV + 7Y + 
dz 
jz * 
7 x dx 
dv= yx 
ydy 
zdz 
div = 
x 2 dx 
xjx 
+ 
y 2 dy z 2 dz 
VF 
We then have 64 ^--functions and an «-function, viz. writing 
and then 
6 = y — z .z — x.x — y, 
Vo, = Vaa^ (8 equations) 
Vabc = ^ {{y — z) Va 1 b 1 c 1 a 2 b 2 c 2 defgh + (z — x) Va 2 b 2 c 2 abcd 1 e 1 f 1 g 1 h 1 + (x 
y) FabcajbiCidiAjfigaha} 
(56 equations)