VARIATION. 



If now, as in the fecond problem, we make 



d V = M d *■ -h N d^ + P d/> + Q d y + &c. 



we (hall have 



dicJV-dV^.v = Ndx(Jj-/)Ja:)+Pdx(J/--?^^) 



+ Qd;r(^7-rJx) + Rd*(Jr-x^>:) + &c.) 



VZd*- = ZdAr+/ |N"dx( 



An equation which, being of the fame kind as Equation (B 

 ij — p^K = ^'w, and fuppofing Ax conftant, 



f 



Subftituting, in equation ( K), inftead of/L'd a- (d x J V - 

 dV^x), its aaual vJue/i' (d;r ^V - d V Jx), and inftead 

 ofd;r5V — dV^x, tlie value affumed aDove, reuniting the 

 feveral parts, and for the fake of abridging, making ^' N + 

 N' = N", i' P + P' =r P", /f' Q + Q' = Q" &c. this equation 

 will become 



y^pix) + P"dx (J/) 



- r} x) 



q^x) 



8cc. 



+ K"Ax['^r-s^x) + 

 + /P"dx [Ip-q^x] 

 + /R"dx [Ir-six) + &c. 

 Prob. II. will give in the fame manner, by making 



ifZA: 





fixStu 



+ Z^x + 



+ 



, dP" 



+ 



d' Q" d' R" 



[P"- 



d J w 



"dT 



dMw 

 Ax''' 



dQ" d_ 

 dx ■*" dx 

 d R" 



d^ 

 R" 



+ &c. 



- &c. ) 



(R" - &c.) + &c. 



(L) 



It may be remarked here, the fame as in Problem II., that Now let 5jr - /. J a; = J w, and J z - /.' ^ x = 5 w'; and 

 the expreflion of this variation includes two diftinft fpecies fuppofing dx conftant, we (hall have, by precifely fimilar 



of terms ; viz. thofe which are affefted with the fign/, and 

 thofe that are free from it ; and moreover, that the in- 

 tegrating by parts introduces certain conftant quantities, 

 which are additive to the terms of the fecond fpecies ; and 

 that the aggregate of the terms afFefted with the fignyj ex- 

 tends through all the variation, viz. from the place where 

 it commences to that where it finifhes ; while the other terms 

 anfwer only to the beginning and end of the variation. 



Prob. V. 



To determine the variation of the indefinite fimple inte- 

 gral y"V d X, where V is any given fundion of three variables 

 X, y, and 2, and their differentials. 



We Ihall have at firft, the fame as in the formula of t-wo 

 variables, 



J/Vdx =V;?x-/(dx^V -dVJx) (M) 



Let us fuppofe A y = pAx, A p-=. qAx, A q ^= r A x, 

 d r = J d X, &c. d z = /)' d a:, d p' = y' d X, d 9' = j" d X, &c. 

 the letters/", q, r, s, &c. p', q', r', &c. cxpreffmg funftions 

 of X, y, z, and their differentials. 



Now making 



jjY_fMdx-l-Ndj>+Pd/.-l-Qdj-l-Rdr-(-&c. 

 1 -I- F d z + G d/.' + H d j' -1- I d r' -I- &c. 

 and hence, alfo, 



J Y_ fMJ^-t-NJji-f PJ/>-f-QJj+ RJr+&c. 

 X+YSx + G^p' + n^q' + lSr'+ &c. 



expreffions in which N, M, P, Q, R, &c. F, G, H, I, &c. 

 are given funftions of x, y, z, />, q, r, &c. p', q', r', &c. we 

 ihall find 



AxSV -AVix = 



HAx {iy- p^x) + PAx [S p- yJx) 

 + QAx {iq- rix) + R d.r (Jr - s S x) + &C. 

 + F d^(Jz-/)'Jx) + G d.r {Sp' - q'Sx) 

 -l-Hd«(Jg'-r'Sx) + I Axi^rl -s'Sx) + &c. 



operations to thofe performed in Prob. II 

 S/Y d X = 



dP d'Q AiR 



/Ax^'ivlN 



+ 



Ax Ax" 



dx' 



-|-/d;>:J«;'(F 



-f VJ* -f St 



dG 



dx 



+ 



d'H 



+ 



+ 

 + 

 + 



+ 



«>(P - 

 Ax \^ 



d;c "^ d 



d R 



~ "d7 



+ 



+ &c.) 



■ d^ + ^^-l 



^-&c.) 

 x' > 



&C-) 



d'Si 



dx' 

 &c. 



d^ 



(R - &c.) 

 &c. 



' ( I - &c.) 

 &c. 



+ 



d»I 



- &c.) 



-1- &c. 



' dx^ 



4- &c. 



To which it will be neceffary to add certain terms, in order 

 to complete the integral, as ftated in the conclufion of our 

 fecond and laft problem. 



The formulae above confidered are the fimpleft of their 

 kind, and the folution of them is found by a calculation com- 

 paratively direft and eafy to perform ; but it may happen, 

 that in the general expreflion y'Z Ax, of which the variation 

 is required, the quantity Z is a funftion of many variables, 

 confifting of algebraical expreffions and various indefinite 

 fimple integrals ; or the quantity Z may depend upon the 

 integration of an equation of any order ; it may alfo, in fome 

 cafes, be required to find the variation of a formula under 



a double 



