174. If M be a function of two variables, x and y, and
we put the partial differential coeflBcients of u regarded
as a function of x, and u regarded as a function of y re-
spectively equal to zero, and find the values of the varia-
bles from these equations, then, if

dv? ' dy^ \dxdy) '
the second partial differential coefficients having the same
algebraic sign, u will be a maximum when that sign is
negative, and a minimum when that sign is positive.
If the second differential coefficients reduce to zero on
substituting the values of x and y determined by patting

^=0 and 'jZ—O, the function will not be a maximum

or minimum.
For, in Art. 64, u=f(x, y),
and u'—flx+h, y+k) ;

du^ dHi h?
then u -u= +^h+ ^ J-2+, etc.,

du d'^u , ,

d^u k^
+ dy^ 172+' ^^'
If, in Art. 64, we had taken
u=f{x, y),
and u"=f{x—h, y—lc),

we would have had

„ du d'^u h^

" -''=-^'^+ d^ i"^-' ^*°-'
du d'^u

-ITy^+d^ ^^ -' ^*°-'
d'^u ^2

+ d^ lT2~' ^^•
^.du du
If ^ and ^ be of finite values, it is obvious that the

quantities h and k may be taken so small that the first
terms of the second members of w'— m and u" —u may



MAXIMA AND MINIMA OF FUNCTIONS.
233



be greater than the sums of all the terms that follow ; in
which case one of them will be positive and the other
negative, and therefore u can not be a maximum or a
minimum ; hence, by reason of the independence of x
and y, and therefore of h and k, it follows that

dti du

^=0,and^=0.
