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A MEMOIR ON THE DOUBLE ^-FUNCTIONS.
[665
Instead of considering in like manner the radical s/a — x.b—x.c — x, I pass at once
to the radical s/a — x.b — x.c— x.d — x\ and starting from the differential relation
du —
dx
s/a — x.b — x.c — x.d — x'
and using the single letters A, B, C, D, il to denote functions of u, I assume as
definitions
A = H s/a — x,
B = il s/b — x,
C = H s/c — x,
D = O s/d — x,
and another equation to be presently mentioned; A, B, G, D are called ^--functions,
and il is called the ««-function.
But before proceeding further I introduce some locutions which will be useful.
In reference to a given set of squares or products, I use the expression a sum of
squares to denote the sum of all or any of the squares each multiplied by an
arbitrary coefficient; and in like manner a sum of products to denote the sum of
all or any of the products each multiplied by an arbitrary coefficient: in particular,
the set may consist of a single square or product only, and the sum of squares or
products will then denote the single term multiplied by an arbitrary coefficient. In
the present case, we have the quantities s/a — x, s/b — x, s/c — x, s/d — x, and the squares
are a—x, b — x, etc., -which belong all to the same set; but the products (meaning
thereby products of two quantities) fa — x. b — x, etc., are considered as being each of
them a set by itself. A sum of squares is thus a linear function X + px, and
conversely any such function is a sum of squares; a sum of products means a single
term z> fa — x b — x (or vs/a — x.c — x, etc., as the case may be), and conversely any
such function is a sum of products: the coefficients X, p, v may depend upon or
contain il, and in differential expressions (du being therein considered constant) the
coefficients X, ¡x, v may contain the factor du or (du)-—and if convenient we may of
course express such factor by writing the coefficients in the form X du, or X (du) 2
etc., as the case may be.
We may now explain very simply the form, as well of the algebraical relations,
as of the differential relations of the first and second orders respectively, which
connect the functions A, B, G, D.
The functions A 2 , B 2 , G 2 , D 2 are each of them a sum of squares, and hence there
exists a linear relation between any three of these squares. But the products AB,
AG, etc., are each of them a sum of products (meaning thereby a single term, as
already explained); and hence there is not any linear relation between these products.
Considering the first derived functions dA, dB, etc., these each contain a term
in dfl, which however disappears (as is obvious) from the combinations AdB — BdA,
