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 u 
=x +av, co , whose differential putting du=o, dv=o, 
gives c=V ay. But this simpkcity could not be obtained if 
v had been of a more complicated form, as, for example, 
if u=x*-+azxy, as above, and1=2*y?+3x7y5. This does 
not come under Professor Fsher’s form, but the substitu- 
tion of y=a7t makes u=2° sep ae v= '2(t? +3), 
whence m=3,n= 12,2", T=— SABE , whose dif- 
~ (4 38*)2 
ferential “auld ie fy mA thence x 
Various Substitutions may be made hesidas those we 
have used; as for example, y=x?t, y?=x%t, y=a? t+ 
ex*t,&e. Andif by ary of these, or other similar substi- 
tatiana: we can reduce u and v to the forms u=U. T’, 
v=V. T”, U and V being fonctions of one of fl the unknown 
quantities, (as for example 2) and T’, T” functions of t, 
we may from the last, Y= find x equal to a func- 
tion of >" Substituting this in U, we get u equal toa 
: 2 
function of Tr 
putting du=o and do=o, we obtain the value of t, and by this 
means, in many cases, we may solve the problems, ina very 
simple manner. itis unnecessary to enter into any greater 
de: ee hams we hive said will explain the principles of the 
B. 
andé, or ofv and ¢, Taking the differential, 
eth 
oe April 18, 1824. 
Remark by the Editor. 
The above papers, originally intended for the Boston 
Journal of Plilosophy and the Arts—have been throngh 
the candour of the Editors of that Journal, and with the 
oe of the author, transmitted for insertion in this 
wo 
