655] A MEMOIR ON DIFFERENTIAL EQUATIONS. 97 
and, adding for greater distinctness the next following cases, 
12345 = 1.2345 + 2.3451 + 3.4512 + 4.5123 + 5.1234, 
123456 = 12.3456 + 13.4561 + 14.5612 + 15.6123 + 16.2345, 
where of course 2345, &c., have the significations mentioned above. 
Dependency of Functions. Art. Nos. 7 and 8. 
7. Two or more functions of the same variables may be independent, or else 
dependent or connected; viz. in the latter case any one of the functions is a function 
of the others a = a (sc), b = b (sc), the functions a, b are dependent, but if 
a=a(x, y), b = b(sc, y), 
then the condition of dependency is 
d(a, b) 
= 0, 
d(x, y) 
and, similarly, if a=a(x, y, z), b = b(x, y, z), then the conditions of dependency are 
d (a, b) 
d (x, y, z) 
= 0, 
viz. if the equations thus represented are all of them satisfied, the functions are 
dependent, but if not, then they are independent. 
Observe that, when a=a(x, y, z), b = b(x, y, z) as above, if we choose to attend 
only to the variables x, y, treating z as a mere constant, there is then a single condition 
d(a, b) 
of dependency ^ = 0, and so if we attend only to the variable x, treating y, z as 
mere constants, then a and b are dependent. Thus when a = x, b = x 2 + y, the functions 
a, b are independent if we attend to both the variables x, y\ dependent if y be 
regarded as a constant. 
8. Further when a — a(x, y), b = b(x, y), c = c(x, y), the functions a, b, c are 
dependent; but when a = a (x, y, z), b = b (x, y, z), c = c (x, y, z), the condition of depen 
dency is 
d (a, b, c) 
d(x, y, z) 
= 0: 
and so when a = a(x, y, z, w), b = b (x, y, z, w), c = c (x, y, z, w), the conditions of 
dependency are 
d(a, b, c) 
d (x, y, z, w) 
= 0; 
viz. if all the equations thus represented are satisfied, the functions are dependent; 
but if not, then they are independent. And so in other cases. 
c. x. 13 
13