665] 
A MEMOIR ON THE DOUBLE ^-FUNCTIONS. 
191 
case may be) of these differentials; thus, in speaking of AdB — BdA as a sum of 
products, it is implied that the coefficients of the several products are linear functions 
of du, dv, and so in speaking of A d 2 A — (dA)- as a sum of squares, it is in like 
manner implied that the coefficients of the several squares are quadric functions of 
du, dv. 
An ¿^/-function is simplex, such as \Ja, or complex, such as Vab ; the square of 
the former is aa! = a 2 — a (oc 4- y) + xy, which is of the form \ + p, (x + y) + vxy ; the 
square of the latter is 
= ^ {abfcjdjej + a^fjcde — 2 VX Y}, 
where observe that the irrational part 
2 
6- 
VXF 
is the same for all these squares : 
so that, taking any two such squares, their difference is = ^ multiplied by a rational 
function of xy: this rational function in fact divides by 6'\ the quotient being a rational 
and integral function of the foregoing form X + p (x + y) + v xy. Hence selecting any one 
of the complex functions, say \/de, the square of any other of the complex functions 
is equal to the square of this plus a term X -f- p (x +y) + v xy ; and hence the square 
of any function simplex or complex is of the form X + p (x + y) H- v xy 4- p (\/de) 2 ; this 
being so, the squares of the .ry-functions may be regarded as forming a single set ; 
every sum of squares is a function of this form X + p(x + y) + vxy + p (Vde) 2 ; and 
conversely every function of this form is a sum of squares. A sum of squares thus 
depends upon four arbitrary coefficients X, p, v, p ; and we may, in an infinity of 
ways, select four out of the 16 squares such that every sum of squares can be 
represented as a sum of these four squares each multiplied by the proper coefficient ; 
say as a sum of the selected four squares : in particular, each of the remaining 
squares can be expressed as a sum of the selected four squares. It appears, by the 
first of my papers above referred to, that there are systems of four squares connected 
together by a linear equation : we are not here concerned with such systems ; only 
of course the four selected squares must not belong to such a system. 
We have the products of the ¿cy-functions, where by product is meant a product 
of two functions. The number of products is of course = 120, but distinguishing these 
according to the radicals which they respectively contain, they form 30 different sets. 
Thus we have 
Vb 'Jab = ^ {b VafbjCjdA — Vaffjbcde}, 
c i » }> 
dj „ }, 
Gj ,, ],