138 Remarks on Several Subjects. 
I have solved this problem as an example of the general 
method, but it is very evident that a simple solution might be 
obtained by substituting y= in u, by which it becomes 
: x ll 
— 9 be e e . 
=xr>+av.x  , whose differential putting du=o, dvu=o, 
gives c=V ay. But this simplicity could not be obtained if 
» had been of a more complicated form, as, for example, 
if u=x*+azy, as above, and v=x*y?+3x7y5. This does 
not come under Professor Fisher’s form, but the substitu- 
tion of y=x?t makes u=ax? (1-+at), v=x2'?(t?+3¢5), 
whence m=3, ee Lp ne » whose dif- 
soi a a> | onze aes) 
ferential put =o, would give t, and thence a, y. 
Various substitutions: may be made besides those we 
have used; as for example, y=a?t, y?=x7t, y=a?t+ 
ex*t, &c. Andif by any of these, or other similar substi- 
tutions, we can reduce wu and v to the forms u=U. T’, 
v=V. T", U and V being fnnctions of one of the unknown 
quantities, (as for example x) and T’, T” functions of ¢, 
we may from the last, Va find « equal to a func- 
Vv 
TT" 
tion of 
. Substituting this in U, we get u equal to a 
function of Tm and ¢,orofv and?t. Taking the differential, 
putting du=o and dv=o, we obtain the value of ¢,and by this 
means, in many cases, we may solve the problems, ina very 
simple manner. [tis unnecessary to enter into any greater 
detail, what we have said will explain the principles of the 
method. 
Boston, April 18, 1824. 
Remark by the Editor. 
The above papers, originally intended for the Boston 
Journal of Philosophy and the Arts—have been through 
the candour of the Editors of that Journal, and with the 
consent of the author, transmitted for insertion in this 
work. 
