186 
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,