of a Function of any number of Variables. A17 
i remeber sisal oy valley 
dg 8 iggy maple Ne 
d du du du 
(su 7 oS ag hone ™ tz —ke.); 
f du 
where w’ is put for 7. 
19 dy du du 5 ‘ ae ae ; 
ow = x 4- ay dy+ mo z+&c. is the variation in the fune- 
tion w, that is due to the variations dz, Sy, dz, &c. independently 
of any change in the form of that function. And the excess of 
the whole variation 6u above this partial variation, may 
garded as the variation occasioned in the function by its change 
of form. Let this excess be denoted by Ju, and let a similar 
notation be used in other like cases. Then 4u'=~4—; that is, 
du dau 
naa? a Hence we readily derive the general equation 
9 qitmtr&e.4, qitmtr&e. 4, 
(2) “deldy"dz’&e.  da'dy"dz"&e. | 
Now for brevity, let D be put for the complex characteristic 
q’t™+r&e. 
daldy"dz"&c. } then since 4Du is only a substitute for the ex- 
Te dDu dDu. dDu 
pression sDu—-~7- or. = TdyOat da 
bis : du du ree du jt-8s 
: n proved,is equal to D.4u ot D(du—5, Sry oo is z+ e.) 
it follows that 
dz —éc. and, as has just 
du du du 
(3.) 'Du=D(du- 7, Gy by — 5. dz—&e.] 
dDu dDu dDu 
+e wha dy a dz ae 
which is the general formula required: for it exhibits the value 
of the variation of any differential coefficient of u, due to the 
Ssumed variations du, dz, dy, 9z, &c. in terms of those varia- 
tons ; in other words, it assigns for the dependent variation Du, 
in which 9 is separated from u by the characteristic D, an expres- 
sion in which none but the assumed variations occur, in which 
the characteristic 6 is found, only as émmediately preceding one 
or another of the variables u, x, y, 2; §¢- va re 
Ve have now to state a second method of obtaining the same 
result; the method to be preferred as the most simple and direct. 
_ We will begin by showing that when a function varies in con- 
Srconp Series, Vol. II, No. 9.—May, 1847. 53 
