VARIATION. 



This being premifed, and making here for the vahies of 

 d V and i' V the fame fuppofitions, and the fame calculations 

 as in the preceding problem, we (hall obtain 



_,,,., dP d'Q d3R 



hS' 



d Q d»R 



dx^ dx' 

 — &c.) 



(f-=^ + 



Ad^iy,_ dR \ 



^ (Q - Tx + ^^•) 



d 



hd^^ 

 dx' 



(R-&C.) (G) 



And we (hall find in the fame manner, by putting /; N for 

 N, /!. P for P, * Q for Q, &c. 



/■/^ (dxJV- dV^x) - 



fdx 



(,N-^4^+- 



d X 

 dx 



+ ^ TO (/^ P - 



d^TO,,^ d(ARl ^ 



d'iw 

 + ^j^(/.R-&c.) 



^fZdx- 



Finally, fubftituting in Equation (G), inftead of 

 hf [dxiV - dVix) ^nd/i [dxSV - dV^A;), their 

 values, which we have found above, we fhall have the ex- 

 preffion for the variation of iyZ dx. 



As in thefe forts of problems, it is required to find the 

 variation which anfwers to a given abfcifs a, it is evident 

 that in denoting by H the integral fT d x, correfponding 

 to this abfcifs, we may regard H as a given conftant quantity 

 relative to the total variation, while h condantly reprefents 

 the indefinite integral /T d x, that is to fay, the integral for 

 an indeterminate part of the abfcifs a. 



Then, in writing H for h in the part 



i/(dxiV -dV^x), 



and pafling H under the fign of integration, the expreflion 



V(d.v^V- dV^v) ~/f,{dxSV-dVSx) 



will become 



/H(d.v^V-dV^«) -/h{dxSV-dVSx) = 



/(H-A) (dxiV-dVJx). 



If, for the fake of abridging, we make H — k =: i, which 

 gives d{H — />) = d/f, H being conitant : now, making 

 conformably to thefe remarks, and to thefe abbrevia- 

 tions, the fubftitutions indicated at the end of the preceding 

 article for equation G, we (hall find that tliis equation 

 becomes 



/d.^^(iN-li^ + qi^-^-iW_&c.) 

 ^ dx dx' d x' ' 



+ ZSx + iSw{iF-^^-^ + &c.) 



idS^f,^ d(/-Q) 



d; 

 id' 



[iQ 



+ Sec.) 



dx' 



{iK- &c.) 



(H) 



Pbob. IV. 



To determine the variation of the indefinite integral for- 

 I mula/Z d x, Z being a funftion of x, y, p, q, r, &c. ; and 

 I of the indefinite fimple integral formukyV d x, where V is 

 I the fame as before. 



Firft, we have 



l/Zdx = 7.lx -\-f[dxhZ-dZlx) (I) 



Let us fuppofe /Vdx = /, orVds;=d/. The quan- 

 tity Z being given in /, x,y, p, q, &c. we (hall have 



dZ = L'd< + M'dx + N'djr+ P'd/.+ Q'dy + &c. 

 iZ=L'J< + M'^* + N'Jji + P'^/. + Q'^9+ &c. 



expreflions in which the quantities L', M', &c. are funftions 

 of/, X, y,p, q, Sec. 



Whence we draw, by a proceeding fimilar to that which 

 has been employed in the fecond problem, 



d^JZ - dZix= L'{dxh-dtSx) + }i'dx{iy-pSx) 



+ P'd.v [Sp-qix] + Q'dx{Sq-rix) 



+ R'dx{Sr-s^x) + &c. 



Butd/ = Vdx,andh=zS/Vdx = Wix +/{dx!>V 

 — dV Ja-) ; therefore 



L'(d^^/-d/J.x) =L»d^(d^^V-dV^*); 



and thus Equation I. becomes 



^/Zdx=:Z^x +/ ^L'dxf{dxSW - dV5x)| 



+ /[^'dy/(Sy-pSx)'^ 



+ 'P'dx{Sp-qSx) + Q>dx (Sq-rSx) 



+ Rldx{Sr-sSx) .+ &c (K) 



Now if we reprefent by A' the integral fUdx, and 

 take, by parts, the integration of the fecond term, we 

 (hall have 



fL'dxf {dxlV -dV^x) = 

 h'f{dx^V-dVlx) -fh'[dx^V-dV^x). 



Let UB fuppofe now that the value of the integral/ L' d x, 

 for any determinate abfcifs a, is H', and that h' ftill con- 

 tinues to reprefent the indeterminate integral/L'd x; then, 

 by pafling H under the fign of integration, the tenn 



/ JL'd.v/(dx-^V- dV^x)| 



becomes 



/(H'-//) [dxlV -dVlx): 



or \in'-h' = i, then 



fki {dxlV -AW^x). 



4K 2 If 



