VARIATION. 



(.V + i x) for X, and {y + ci'jr) for y, and the equation 

 becomes 



{y + hY^{''+^''){^ + ^'=)i 



from which fubtrafting jr ^ = a k, we (hall have 



2ySy^^xia-\-aSx; 



Sa + a Sx X S a + aSx 



therefore, 



J^ = ' 



2y 



which is an expreflion for the variation s ri, of the ordi- 

 nate P M. 



In this example, (and the fame has place for all fimilar 

 equations,) the parameter a, and its variation S a, are con- 

 ftant quantities for the entire parabolas, while thofe of the 

 co-ordinates P M and A P continually change ; the changes, 

 therefore, relative to the fame parabola belong to the dif- 

 ferential calculus, and thofe which refult from the paffage 

 of one parabola to another, to the calculus of variations. 

 Any one of the variations S <j, S x, ^y, may be arbitrarily 

 aflTumed ; as, for example, we may fuppofe ^ x =: d x, but 

 this fuppofition being once made, the values of the other 

 variations muft be fubordinate to this, and we cannot there- 

 fore afterwards make ^ y :^ Ay, or oy z=^ a. 



There is no difficulty in determining the variations of 

 every order for algebraical and circular quantities, and 

 common exponentials ; the operations being exaftly the fame 

 as in the differential calculus ; we therefore obtain the va- 

 riations by the fame rules, and have only to write ^ inftead 

 of d, and in this refpeft the caulculus of variations return 

 again to'the differential calculus ; but this latter will not be 

 fufficient when it is required to determine the variation of 

 formulae, which contain in themfelves the fign of inte- 

 gration : thus, for example, let the integral formula be 

 f V ix, where V is any funttion of x, y, and z, and con- 

 ftant quantities; we difference this by omitting the figny"; 

 that is, d (/ V d X ) = V d x ; but the exprelTion ^/ V d x is 

 very different, as we fhall fee in what follows. 



Now the principal objeft of the calculus of variations, is 

 to determine the variation of thefe forts of integral formula ; 

 let us, therefore, endeavour to eftablifh the principles which 

 are to ferve as the bafis of this refearch. 



Firjl Principle. — The variation of a differential is equal 

 to the differential of a variation, and reciprocally ; that is, 

 we fhall have Sdz- = d.5r. 



For let us fuppofe that the variable t reprefent the or- 

 dinate of a curve ; then this ordinate will change by differ- 

 entials while it belongs to the fame curve, and by -variations 

 in palling from the propofed curve to the curve indefinitely 

 near to the firil. In the primitive curve, let ir' be the confecu- 

 tive value to v, and confequently w' = i- -f d x, or d t = 

 ■k' — TT. Now taking the variation of this lafl equation, we 

 fhall have dd7r=:^x' — } v ; and in the fame manner as t 

 and tt' are confecutive values in the feries of t's, we may 

 confider St and ^!r' as confecutive values in the feries of 

 Jtt's, fothatK' = Jn- -I- dK, or d^TT : 

 in equating thefe two values of ^ ir' — 

 ^ d X = d J T. 



Hence if we have an expreffion which contains any num- 

 ber of d's and 5's affefting one and the fame variable, we 

 make thefe charafterillics change place at pleafure ; for we 

 have feen that J d tt = d J x ; and in the fame manner, we 

 may for J d' t write d i d x, or d' ^ t ; and for }> A' v we 

 may write d ^ d^ w, or d M d r, or d ^ J t, and fo on of 

 others. 



Second Principle The variation of an integral formula is 



«qual to the variation of its differential; that is, J/j =/J !• 



Vol. XXXVI. 



■' — J X ; thus 

 we fhall have 



Let /I = 2, and confequently f = d a;, we fliall have by 

 taking the variations ^|=;5d2, or5| = d52; and inte- 

 grating this laft equation, we obtain /i' i =o%— J/|. 



Hence in repeated integrations, we change at pleafure 



the figns/and^.; for we have feen that .5/| = /^ J ; and 



in the fame manner, ^/l = /^/f =ffH; fo alfo 



V//I = P/A = //Vl =///H; and fo on of 

 others. 



On the Method of determining the Variations of indefinite in- 

 tegral Formula:. — By indefinite integral formulae, is here to 

 be underftood thofe exprefTions which contain the fign/, 

 and fuch at the fame time that the integration cannot be 

 effedled : thefe formulae are faid to hefimple, when they con- 

 tain only one fignf, and compounded, when they contain two 

 or more fuch figns, or when they are any function of fimple 

 integral formula:, combined or not with algebraical quan- 

 tities, by addition, multiphcation, or divifion. 



Let us begin by confidering thofe formulx which contain 

 only two variable quantities x and y, and between which we 

 fhall always fuppofe the relation dy = ^ d x, dp = q d x, 

 dq := rdx, d r =^ j d x, &c. a fuppofition which it will be 

 very neceifary to bear in mind. 



Problem I. 



To invefTiigate a general rule for determining the variation 

 of any indefinite integral formulayTrd x. 



Whatever may be the quantity a-, we have always from 

 the fecond principle given above, Ifvdx = f^ (^-dx); 

 butS(Td.v) = dxoTT 4- ir^dx; and the firft principle 

 gives J d X = d ^ X ; whence ^/tt dx =fdx^ir 4-/wdJx. 

 Now by the method of integrating by parts, the laft term 



f- dS X := Trd.x — yd ttSx; 



whence by fubftitution, 



Sfirdx = X Jx + yd X J TT — ydff Jx, or 



Jy^rdx = !r^x +f{ dx^T — dvt^x) 



Now the different values that we may attribute to a-, will 

 give rife to different general problems ; of which we fhall 

 develope a few of mofl common ufe, and which will open 

 the way to others of a higher kind. 



Prob. II. 



To determine the variation of the indefinite fimple inte- 

 gral yV d X, V being a given function of .v, y, %, p, q, r. Sec. 

 Firft, by the preceding problem we have 



VVdx = Vdx -I- y(dxiV - dV^x) .... (A) 



Again, the quantity V being a funftion of x, y, z, &c. 

 we fhall have, by taking the differentials and the variations, 

 the two equations 



dV = Mdx + Ndjr-hPd;)4-Qd9+Rdr+&c. 



S V = M ^x + N d j; + P ^V + Q ^ ? + R ^ »■ + &c. 



in which the co-efficients M, N, P, Q, R, &c. (which are 

 the fame for both equations] reprefent given funftions of 

 X, y, p, q, r. 



Multiply the firft of thefe equations by ^ x, the fecond 

 by d X, and fubtraft the firft produft from the fecond, and 

 we fhall have 



dxlV - dVo.v = N [dx^y — dylx) 



+ P [dx^p - dpSx) 



+ Q{dxSq - dqLx) 



+ R(dxJ*- - drix) 



4K. If 



