134 Remarks on Several Subjects. 
quantities x and y, and so doponta* on each other, that # 
is to be a maximum or minimum when v isa given quantity. 
The most natural way to vali such problems, is to find the 
value of one of the unknown quantities, as for exainple, y, 
in terms of «x, by means of the given quantity v. 
tuting this in the function u, it becomes a function of the 
other variable quantity x (independent of y ;) and its differ- 
ential being taken relative to 2 and put =o, will give the 
maximum or ininimum of u, according to well known prin- 
ciples. The same result would be obtained, if we find 
from v in terms of y, and substitute it in u, by which means 
it will become a function of the single variable quantity y, 
(independant of «) whose differential relative to y put =0, 
will also give the maximum or minimum of u. As either 
of these methods will answer, it will be in aur power to use 
that which leads to the most simple results ; but some- 
times the function v is of so complicated a form, that it is 
difficult to determine the value of x or y, and some anal- 
ytical artifice must be used to obtain the required solution. 
of these artifices consists in the introduction of a new 
variable quantity ¢ instead of cory. Thus if the function 
u is a homogeneous expression in x and y of the order m, 
or such that the sum of the exponents of x andy in every 
term of u ts exactly equal to m, we may, by the substitution 
of y=<axt reduce it to the form u=z2". T’, T’ being a 
function of ¢ exclusive of «x,y, andthe same tps: of 
=«ct inv, supposed to be a homogeneous function 
the order n, would reduce it to v=a2", T’, T/ betip snails 
er function of t independent of x, y. Now the value of « 
found _ this last expression and substituted in w, will 
give — p= =P ~ the second member of this equation be- 
ing a function of ¢ alone ; i and, if for simplicity we pul 
t= 0 (=") 
re v 
it will become T=—. Taking now the differential of this 
equation, the iit) “member will vanish, when wis a max- 
imum or minimum, because then du=o, and (v being con- 
