VARIATION. 



a double or triple, Sec. fign of integration, as //7.dxdy, 

 in which Z is any funftion of x and y, and fo of others. In 

 all thefe cafes, except the laft, the variations are detern?ined 

 in the fame manner, but the calculus of courfe becomes 

 more long and intricate, which our limits will not allow of 

 our entering upon in this place. On this head, therefore, 

 the reader is referred to the feveral works mentioned in the 

 introduftion to the prefent article. We only propofe giving 

 here one problem, by way of illuftrating the preceding cal- 

 culus ; -viz. 



To determine the curve OMD (Plate Kill. fg. 2.) 

 through which a body will pafs from the point O to D, not 

 in the fame vertical line, in the (horteft. time poffible. 



Let A V reprefent the vertical plane, in which are fituated 

 the two given pokits O and D ; A V the axis of the abfcifs ; 

 and the horizontal line A F that of the ordinates. Alfo, 

 let us fuppofe any abfcifs A P = x, the ordinate P M = y, 

 and confequently the element of the arc M m = ^ (d s' 

 i + d j;") — d .v y ( I +/"')> making d y = p d x. 



Now whatever may be the nature of the curve OMD, 

 the velocity of the body along and in the direftion of the 

 element of the curve M m, is equal to that which it would 

 have acquired in falling from a certain vertical height j all 

 thefe heights deriving their origin in the fame horizontal 

 line, which we may fuppofe to be the axis of the ordinates 

 A Z, the pofition of this axis being arbitrary. 



Thus, calling g the gravity of the body, the velocity 

 along M m will be exprefled by ^2 g x, and confequently 

 ' - M _dx ^/{i + f) 



quently, if the whole is equal to zero, thefe two parts are 

 each alfo equal to zero ; thu5 the equation ifVdx-=zo, 

 gives in general the two following equations, of which 

 the one is definite, and the other indefinite ; viz. 



(Oo = /dx5^(N-^-;-'^-&c.) 



r 



(2)0 = 



V^ 



+ I IV ( P 



d X d ;■ 



R 



+ 



d^. 



d.'^ 



'(« 



d R . \ 



.(R 



dx 

 &.C.) 



' dx' 

 -h &c. 

 + C, correftion. 



Equation { i ) is that on which depends the nature of 

 curves, fince the fecond member of this equation is an inde- 

 terminate expreffion, which being made equal o, gives to 

 the curve OMD the charafter of a maximum or a minimum. 

 As to Equation (2), it belongs only to the extreme points 

 of the curve OMD, which may be fubjeft to particular 

 conditions, wholly independent of the nature of the curve. 



Now differencing Equation ( i ), and dividing the whole 

 by d K ^ w, we fhall have 



dP d'O d3R 



ax Q , 



+ &c. 



the time in paffing M ;n 

 therefore we (hall have 



V^g " 



V 2<fX 



fimply 



/ 



V 2g X 

 d x y ( I + /.') 



^ a mmtmum, 



which gives generally the fdlution of the problem, where 

 only the nature of the curve is required, that renders 

 fVdx a maximum, or a minimum ; V being a funftfon of 

 the perpendicular co-ordinates x and y of the curve, and of 

 the quantities p, q, r. Sec. which are given by the hypo- 

 thefis djj — p dx, dp =: q d x, &c. remembering that the 

 differential d x has been fuppofed conftant. 



Now to apply thefe general refults to our problem ; fince 



= a minimum. 



N 



and P = 



P 



Now generally, when a quantity becomes a maximum or a 

 minimum, its variation is equal to zero ; confequently we 

 (hall have 



R = 



'f 



d X ^ (l -I- j>^) 



^x. V(I +/' 



&c. our Equation (3) becomes 



■Z2^( P 



; alfo Q = o, 



o = 



k.V * 



Now this agrees with our formula fVdx in the fe- 



o=d(- 



V (I +P' 



). 



cond problem ; v'fz. in the prefent cafe V = 



V" * 



confequently we fliall have d V = — 



2 X . / X 



« + 



which is the differential equation of the curve OMD; con- 



P 



fequently, by integrating we fliall have 



a/ X . 



• dp; an expreflion which, being com- 



V a 



a being an arbitrary conftant quantity. Now 



V»- V(i +/) 



pared with the general value 



dV = Mdx -f-Nd;'-t-Pd/.+ 

 gives here 



M=Z_.^ii-L^;N = o = 



&c. 



p = 



2 X V* 

 P 



v'x. vCn-/-')' 



Q = o, R = o, &c. 



Now the expreffion of the variation i/V d x comprehends 

 generally, as we have feen in Equation (D), two parts, the 

 one indefinite, containing the fign /, and the other definite, in 

 which that fign is not found ; and it is evident that thefe two 

 parts are wholly independent of each other ; and confe- 



a y = a X a/ ; 



•' a - St 



the equation of thereverfed cycloid, its bafe being horizontal, 

 and its generating circle having for its diameter the conflant 

 quantity a. 



This equation being integrated, will receive a Second ar- 

 bitrary conftant b ; and we (hall have then, in the final equa- 

 tion, two conftants, a and b, which will be determined from 

 the condition that the cycloid paffes through the two points, 



O and D, 



