VARIATION. 



If now we put ford,/, ip, dq, dr, &:c. their values Whence, by fubftitution, we have 



^ d X, g d a:, r d X, X d X, &c. we (hall find 

 dxiV - dVJx = 

 f Ndx {Sy - pix) -t Pdx (J/. - 9 Jk) + 

 iQdx {^q- rix) + Rdx (Jr - sSx) + &c. 



Confequently our equation (A) becomes 



J/Vdx = Vdx +/|Ndx(»> -/-J-) -t- 



Pdx(J/- j5x) + Qdx (Jg- rJx) + 



Rdx (Jr -x5x) + &c. I = 



VJ«+/Ndx(ijf-/.ix) +/P <!•>■• (-V-?^*) 



+ /Qdx(5y-r5x)+/Rdx(Jr-xlx)....(B) 



This being eftabli{hed, let us mak^ Sy—pox = ^'w,{ a fub- 

 ftitution that mil be employed in what follows, ) and differ- 

 encing, we (hall have 



dJjp — d/^x— />d^x = d^TO. 

 But the formula dy=pdx gives, by taking the varia- 

 tions, 



J d jr = d X 3^ -)- /> i d X, or 



diy = dx^p+pdSx,or 

 dSy—pdSx = dx^p. 



dx^p — dpix = dSw. 

 Now putting for dp its equivalent j d x, we ftiall have 

 d X d^ — ^ d X d X = d ^ •ai, or 



. . d^ <w 



and as the quantity 5 w has for its value ^ y — pox, by 



And 



_ d[ly-plx] 



dx 



hypothefis we have again ip — q c x 



a calculation exaftly fimilar to the preceding gives alfo, 



Jx = 



dx 



d (i q — r S x) 



dx' 



&c. = &c. 

 Confequently the equation ( B ) becomes 

 J/Vdx = V^x +/Ndx^TO +/Pd Jiy 



Now by the method of integrating by parts, we find, b} 

 making d x conftant. 



/ P d J w = 

 /■ Q d ' 5 TO _ 



P ^ 10 



Q d S w 



f 



dx 



R d' S w 



-f 

 -/■ 



/ d P d* w 

 d Q d ^ TO 



+ 



-/ 





d' Q i TO 



d^ 

 d'^Rd=»TO 



d x^- 



dx' 

 and fo on. 



Whence, making the neceffary fubftitutions, we obtain to the terms of the latter fpecies. The aggregate of the 



fiaally 



r/aw„(N "+0 



)/Vd*= . 



d' R 



-).V^*-f.lx(P-^^^-^ 

 ^ jr d X 



-I-&C.) 



d'R X 



d I 



(Q-'L^ + ^c.) 



-f^(R 

 d X 



+ &c 



d^ 

 &c. 



(D) 



terms affetled with the fign f extends through all the 

 variation, t/z. from its commencement to its termination, 

 while the other quantities anfwer only to the beginning and 

 end of the variation. This remark finds its apphcation in 

 treating of the maxima and minima of quantities. 



Ppob. III. 



To determine the variation of the indefinite compound 

 integral y"Z d z, Z being a given funftion of the indefinite 

 fimpie integral formula yV d x, where again V is a fundlion 

 of X, y, z, />, q, r, &c. as in the preceding problems. 



Firll by Prob. I. 



^/Zdx = Zd";r + /(d.rd'Z - dZ^a-) . . . (El 



1 formula in which dx is fuppofed conftant. 



It will be feen from this expreffion for the variation And fuppofing/V d jr = /, or V d j.- = d / ; finee Z is by 



3/V d X, that it includes two dillinft orders of terms, the hypothefis a funftion of /, we (hall have ^Z = T d /, T 



one affefted with the fign/, and the other free from it. being a given funftion of /, and we (hall thus have J Z =: 



And farther, that the integration by parts necefTanly in- Ti'/; therefore 

 troduces certain conftaiit quantities which muft be annexed dxoZ — dZJx = 'Y dxlt — T dt ix. 



But J/ = VVdx = VJx+/(dx^V-dV^x); 



therefore, d.»5Z - dZ^x := TVdxi* + Tdx/(d.riV - d V i x) - TdtSx 



= Tdx/(d*JV - dVSx]; 

 becaufe TVdxi.v = TdtSx. 



Confequently we have 5/Zdx = Zd'* +/Td*/(dx5V - dVJx) (F) 



Now integrating the laft term by parts, and reprefenting tlie integral /T dx by A, for the fake of abridging, the 

 preceding equation becomes ( F ) 



IJ-Ldx = Zlx + h/{dxiV - d Vix) -/A (d* JV — d Vix). 



This 



