Let V denote fome fi 
fw and y, sand of the ip uaicainy 
fame 
fore become a queftion, amo poffible gai 0 
x and y, to determine that, in which the integral SV 
tween certain limits) is a maximum or minimum ; 
ral when no particular relation ee 
"1s {pecified, oxprefng the meafure of a property belonging 
to all curves, it is required to determine the curve in whic h 
this propercy . a maximum cr minimum. tis evident that 
8.) reprefent this curve, then i in pon other r, 
x muift bea value grea. - 
Gt 
vy & the Pe SVe in the firft cafe 
nd lefs in the fecon d. To fatisfy this ale n, we muit 
-. La pe the difference which any 
e relation of x to y will make in the integral / V 
thi change will be expreffed by making y vary ea 
r in confidering the two curves C E, y+, the fame 
abfeitfa AP correfponds to two ordinates PM, bs rie 
their difference, Mp, fho be diftinguifhed fro 
differences M R, x ¢, which ae = between the aoe 
tive sap a boa on the fame 
ariations as deviled by La eee 
y given aa in 
as varyin o in one 
of anew eee equation in eon the quantities are 
fuppofed to var new relation: and an hypothefis is 
eftablithed upon this fecond operation {uited to the nature 
It is mbol 
curve ; 3 that made in hea tes e af int moving into 
fome new curve: thus M R being reprefented by dy, 
M » will be 395 heres 
/ 
e 
=y + dy; ee) + oy. 
The point M! See to the ee zw’, we fhall have 
=y+dy +3 +4dy) 
=yidy+sy+d 
and by comparing thefe two expreflions of the fame line, 
this remarkable a a is obtained, 
d 
The fame refult ma ie. obta iad without any refer- 
e to the nature of curve lines, by reprefenting by ¢x 
fhe primitive flate of y, aes ave Lx the sage ‘of 
the {uppofed variation. =Jlwe will be 
a certain function of w, a likewife a Aneion of J be- 
caufe of the primitive relation between w and y; let x de- 
note this latter function, then dy = wy; let y + ) = 93 
hen dy! = ry’: hence dy —3 =ary —rysdry 
== oy but dy = y'! — y, taking the eatiations 
ddy= cyl YS hence Sd y = dy. ae? 
a manner 3d*y = d'dy, and generally Jd” y = 
ala it appears ae the fy ‘mbols d and 3 may 
ks be tran d. 
e prir neiple of La Grange’s method confifts in taking 
the differentials of wx, <c. according to the ufual 
procefs, Pa relatively to another charaéteriftic or fymbol 3 
different from d, ufed in the firft cafe; then tranfpofing 3 
after d ay in if it happen to precede them, and making 
Sa, dy, &c. difappear by integration from under the 
fign f. 
Let the formula, aioe within certain oe is to ee a 
maximum or minimum, » a function of x,y, dy 
Suppoting d w conftant, a en 3. fz will be a J al 
value inthe cafe of a maximum or minimum ; therefore 
Vou. XV. 
FUNCTION. 
é. f a de which may be transformed into 
Let the differential ‘of this equation, found in the ufual — 
manner, fi with the fymbol 2, be 
z= Moy + b Nady 4 Pay 4 Bes 
fis +/Nody+/fPoddy + &. = 
Nédy may be transformed into / N d , yy and 
ia. by ee into 
—fd O We 
In the eae manner {Pd 
*dy3 then into Pddy 
of the ies. 
Adding the conftant quantity 4, or correction to thefe 
integrations, the equation becomes 
diy —dPdy + &c. + & 
+f(M—dN+4dP— &c)dy=o. 
As all the differentials of dy have difappeared 
under the fign /, this part is not rae ible of farther 
reduction. Therefore, to verify the equation independently 
of the variations, the co-efficient of dy under the fign / 
muft be made = 0, which gives . following equation, 
T 
— 
ae is transformed firft into 
+ fd 
Poy Py, and fo 
M—dN + 
which equation mult a for ail ce ie of x and x 
containcd within the given limits 
Exa bie — Let it be srequired. to > afl ign the relation be- 
then /da* 4 + dy a minimum, which is 
equiv en to finding die hort Uae between two points, 
3 a ae. " dy: = dxvbdda ad G7 
Yds td Si 
(which, fuppofing ds = VW da* + dy) 
dxwddw dyd3 
ee 
§ 
Here M=o0 N= 
ds 
dy 
na Oy; z= As 
M—dN= —d.5- =o, 
m—- dn = ~a. ao 
Therefore i; = conftant quantity, 
dy 
= = 4, a conftant quantity. 
oo an equation to a ftraight line. 
We ‘hall. conclude this article with Mr. Woodhoutfe’s:me- 
thod of applying thefe principles to the inveftigation of the 
properties of motions, which we particularly wifh to recom- 
mend to the notice of the mathematical reader. 
Let the relation between the {pace 2, and time z, be 
denoted by wx ¢?, then ¢ ci -increafed to ¢ + At, 
wpAn nee gti t.Ar+D'¢ 
+ ke =a + D.x.At+D x. (As) “+ Be. and con- 
ae Av=D.a.Ae4D*, x, (Ar)? + &e, let more- 
ver the relation yal aaa any other {pace, an ri time 
be ea ted by y = then Ay = f (¢ + ng a 
D ft. At+D-ft. ( ara aoe area Dy. 
Cun ae Now, cer Dp 
. D? rh ten the motor of the 
eed ete the {pace Ay, oaches more nearly 
to the motion of the body serine the {pac 
3N 
Fd 
