182 
MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
Since we have 
+ Jxydz, 
if we suppose any two equations to exist between the variables, then y and z are 
functions of x which assume definite values when x is given. Therefore also the 
product y z is a function of x, which may be called <p x. 
If now the two equations are symmetrical, it follows that the letters x, y, z may 
be permuted ; which gives x z = <p y, and xy = <p z ; 
+ z d z 
It is evident that this reasoning may be extended to any number n of variables 
between which there exist n — 1 symmetrical equations, which circumstance renders 
them all similar functions of each other. 
T 
Let r designate their product xy z ... ., and therefore — = y z . . . ., or the pro- 
duct of all except x. 
.*. r + C dx +fjdy +, &c. 
V 
But if — = <p x, we have by merely permuting the letters, 
Therefore 
<Py,T — v z > &c - 
r+C= J*<px.dx-\- J^'Py . dy +, &c.*j 
; 
(A.) 
This equation I first obtained in the year 1821, but not having leisure at that time 
to pursue the subject much further, I contented myself with making a note of it as 
being a subject that deserved to be further examined into. I afterwards found it to 
be the key, as it were, of the whole method. 
In the year 1825 I resumed this investigation, and endeavoured, by the trial of 
various forms of symmetrical equations between the variables, to see whether this 
method would lead to new results, or whether, on the contrary, it would turn out to 
be a mere variation of the methods in common use. 
I here give the results of some of these early trials, just as I find them in the ori- 
ginal papers. 
Ex. 1 . Let the 2 symmetrical equations be 
( 1 .) x + ;/ + z = a 
( 2 .) x 1 -\-y l + z 2 = b 2 , 
a and b 2 being constants. 
