416 On the Variation of a Differential Coefficient 
Let then u=f(z, ae Z, S&C. ), ans ae variation let this equation 
become u= se Ey &c. ):: o let du=u—u, dr=x—z, 
y=y - ened aaa ou —— “Oy, dz, &c. being functions of 
all the independent variables, 2, y, z,&c. These variations being 
assumed, our object is to determine the resulting variations in 
the differential coefficients of w; as for instance, the variation 
gre 
dz 
Mis du du. 
Now by reason. of the variations assumed, a> becomes 7 * 
ie x 
the variation of — = therefore, is — nade and if this be denoted 
dx dx dx 
du -, @u_du_du, 
by 5—, we have the equation 5—-=—-—““: but ice 
y <> we have the equatio i a a ; 
ae — du dow 
cites. dz dx dat dz’ 
how du _dudx du dy du dz 
dx dx dx dy de dz da + ©. 
— er dix dy d(y+dy) aby dz dz 
de de =" Ge de~ de de® de ae 
du diu dadir du diy du diz 
Hence, 6 le de Po da aa —&c. anid if ween. 
=&c. 
du du 
sider that ns ‘a don de’ “and* neglecting the infinitesimal 5>-, sub- 
nitty Sg d he 20D, 6 
stitute dx 1° Fy bt and substitute also —- dy’ a é&c. for =- dy’ yet 
we obtain the equation ‘ 
du diu du dix du diy du ddz 
whe) Agen as ge ga. > 9, gs + a 
This is easily reducible to the following form, which will be 
found convenient for purposes of generalization ; 
ep oH Ou = 6 - _ 
de de ~ ie oni Ws gy 9 HO 
du du yd | 
t dede °C + Tedy Y* Gedz fagihyrester? 
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