A MEMOIR ON THE DOUBLE ^-FUNCTIONS.
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which four expressions form a set, and there are 15 such sets. The set written
down may be called the set af: and the fifteen sets are of course ab, ac, etc.
Again, we have
*Ja*/b= V abajbj,
Vac V6c = ^ {(cid^ + c^de) Vaba^ — (abj + ajb) Vcdelcjdjejf,},
Vad 'Jbd = jp {(dfcjej + djfice) „ - „ „ },
\/ae *fbe = Q- 2 {(efcA + e^cd) „ - „ },
which four expressions form a set, and there are 15 such sets. The set written down
may be called the set aba^j: and the fifteen sets are of course abajbj, acajCj, etc.
The 15 and 15 sets make in all 30 sets as mentioned above.
The expression, a sum of products, means as already explained a sum of products
belonging to the same set; and there are thus 30 forms of a sum of products. The
products of the same set are connected by two linear relations, so that, selecting at
pleasure any two of the products, the other two products can be expressed each of
them as a linear function of these; hence a sum of products contains only two
arbitrary coefficients.
Reverting now to the equations A = X2 \Ja, etc., we see at once the form of the
algebraical equations which connect the 16 ^-functions. Every squared function
A 2 , ..., (AB) 2 , ... is a sum of squares, whence selecting (as may be done in a great
number of ways) four of these squared functions, each of the remaining 12 squares is
a sum of these four squares each multiplied by the proper coefficient; or say it is a
sum of the four selected squares. And in like manner the 120 products of two of
the 16 functions form 30 sets, such that selecting at pleasure two of the set, the
remaining two of the set are each of them a linear function of these.
Considering the first derived functions dA, dB, ..., dAB, ..., each of these contains a
term in 0X2; but 0X2 disappears (as is obvious) from the several combinations IdJ — Jdl
(I write I and similarly J to denote indifferently a single letter A or a double letter
AB): and, without in any wise fixing the value of X2, we in fact find that each of
these expressions is a sum of products.
Passing to the second derived functions, and forming the combinations A d 2 A — (dA) 2 ,
etc., or to include the two cases of the single and the double letter, say Id 2 I — (dI) 2 ,
each of these will contain a multiple of X2 0 2 X2 — (0X2 ) 2 ; but if we assume this expression
il 0 2 X2 — (0X2) 2 = X2 2 iff, where iff is a quadric function of du, dv, the coefficients of
(du) 2 , du dv, (dv) 2 being properly determined functions of xy, then it is found that each
of the expressions in question I d 2 I — (dl) 2 becomes equal to a sum of squares.
It is to be observed that iff is not altogether arbitrary: the equation as con
taining terms in (du) 2 , du dv, and (dv) 2 , in fact represents three partial differential
