134 Remarks on Several Subjects. 
quantities w and y, and so dependant on each other, that u 
is to be a maximum or minimum when v isa given quantity. 
The most natural way to solve such problems, is to find the 
value of one of the unknown quantities, as for example, y, 
in terms of «, by means of the given quantity 7. Substi- 
tuting this in the function u, it becomes a function of the 
other variable quantity x (independent of ys) and its differ- 
ential being taken relative to x and put =o, will give the 
maximum or minimum of u, according to well known prin- 
ciples. ‘The same result weuld be obtained, if we find w 
from v in terms of y, and substitute it in w, by which means 
it will become a function of the single variable quantity y, 
' (independant of x) whose differential relative to y put =o, 
will aiso give the maximum or minimum of uw. As either 
of these methods will answer, it will be i our 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 « or y, and some anal- 
ytical artifice must be used to obtain the required solution. 
One 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 and y in every 
term of u is exactly equal tom, we may, by the substitution 
of y=xt reduce it to the form u=x2". T’, JT’ being a 
function of t exclusive of «,y, andthe same substitution of 
y=ct in v, supposed to be a homogeneous function of x. y. of 
the order n, would reduce it to v=2". T", T’ being anoth- 
er function of ¢t independent of x.y. Now the value of a 
found from this last expression and substituted in u, will 
sir pape sey CH CLAN 
give =P the second member of this equation be- 
ing a function of ¢ alone ; and, if for simplicity we put 
mY Cea 
T’2 \ yt/ 
it will become T=-_. Taking now the differential of this 
mh 
equation, the second member will vanish, when wis a max- 
imum or minimum, because then du=o, and (v veing con- 
