418 On the Variation of a Differential Coefficient 
sequence of changes in the values of its variables and. in its form, 
the whole variation is equal to the variation due to the change in 
the form of the function, independently of any change in the 
variables, added to the variation which is due to the changes in 
the values of the variables independently of any change in the 
form of the function: a theorem which though seemingly ob- 
vious, is not however to be assumed without proof. 
Let u= F(z, y, z, &c.),and by variation let 2, y, z, &. become 
x, y, Zz, &c. while » becomes u+v, where u=Fx, J, 2, &c.) 
and v=f(x, y, z, &c.). Then the whole variation of u which 
we will denote by ou, isu+v-w. But if there were the same 
change as we have supposed in the form of the function, with- 
out any variation in 2, y, z, &c. the variation of wu would be 
fiz, y, z, &e.), which we may denote by v or 4u. And if there 
were no variation in the form of the function w, but the same 
variation as has been supposed in the values of z, y, 2, &¢- the 
, Serta of u would be u- wu, which we will denote by Tv. 
"hen du=4u+Tu+(v—v). But as vis the same function of 
X, y, 2, &c. that v is of 2, y, z, Se. (v— v) is an infinitesimal of 
the second order and may be neglected. We have then finally 
the equation 
(4.) ' du=dutlu; 
a result which verifies the theorem stated above. We now 
proceed to determine the variation of a differential coefficient of 
afunction. Let u be the function, and Du the differential coef- 
du'dy"dz" &e. . Then when uU varies 
qit™*" &e. 
tou-+v, Du varies to Diu+-v), where Dis put for 75 ay" ee 
And if 9Du be put for the variation of Du, 
6Du=D(u+v)-—Du=Du+ Dv— Du. 
But Du is the same function of x, y, z, &c. that Du is of 2, y; 2 
&c.: then agreeably to the notation already used, /Du may 
be put for Du- Du. And as Dy is the same function of x, 
y, z, &c. that Dv is of x, y, z, &c., Dv=Dv+rbv. Hence 
if the infinitesimal Dv, which is of the second order, be neg- 
lected, and if for v, its equivalent 4u be substituted, 
(5.) SDu=D4u+rDu: 
and if for 4 we substitute its equal du— ru, then 
(6.) 6Du=D(du—ru)+rDu. 
Now the meaning of r is such that if infinitesimals of the second 
order be neglected, 
ficient ; where D is put for 
aa du du du 
: (7.) ru=7, ont oy dead dz+&c. 
