456 
ON THE ADDITION OF THE DOUBLE FUNCTIONS. 
[703 
I attach the numbers 1, 2, 3, 4, 5, 6 to the variables x, y, z, w, p, q, respectively: 
and write 
A 12 =Va — x.a — y\ A M = Va — z. a — w ; A m = Va — p . a — q; 
(six equations), 
AB 12 = —— {Va — x ,b — x./ — x. c —y.d—y.e—y — \la — y.b — yf—y.c — x.d — x.e — x)\ etc. 
(ten equations), 
where it is to be borne in mind that AB is an abbreviation for ABF.GDE, and 
so in other cases, the letter F belonging always to the expressed duad: there are 
thus in all the sixteen functions A, B, G, D, E, F, AB, AG, AD, AE, BG, BD, 
BE, GD, GE, DE, these being functions of x and y, of z and w, and of p and q, 
according as the suffix is 12, 34, or 56. 
It is to be shown that the 16 functions A 5e , AB X of p and q can be by means 
of the given equations expressed as proportional to rational and integral functions of 
the 16 functions A 12 , AB 12 , A. u , AB U of x and y, and of z and w respectively: and 
it is clear that in so expressing them we have in effect the solution of the problem 
of the addition of the double ^-functions. 
I use when convenient the abbreviated notations 
we have of course 
a — x = a 1} a — y = a 2 , etc., 
b — x = b 1 , etc., 
d 12 = x- y, 6 U = z-w, 6 x =p-q; 
X — ajbiCjdjejfj, 
A-,« — 
AB 12 = {VaibfficAea - etc. 
1/12 
Proceeding to the investigation, the equations between the variables are obviously 
those obtained by the elimination of the arbitrary multipliers a, /3, 7, 8, e from the 
six equations obtained from 
a0 3 + ¡36* + yd + 8 = e V®, 
b} r writing therein for 6 the values x, y, z, w, p, q successively; we may consider 
the four equations 
ax 3 -f- fix'- + yx + 8 = e VX, 
ay 3 + fiy 2 +7 y + 8 = e V Y, 
olz 3 + fiz 2 +72 + 8 = e Z, 
olvj 3 + fiw 2 + 7 w + 8 = e V W,