the Calculus of Variations. 327 



dec 5, da _ . />*" dV - /**" dV 



l^fdx^ + ly r fdx IT- + &c. + U„ + V /y 8^ + 



(tot-i^) (P.-Q//+&C.} + {V«-?,M} {Qw-&c.} +&C. 



and therefore subtracting this result from the former, we find 



/*/ ^ v P x ' d V y* 1 ' <Z V 



dx — -fS^J rf* — - + **^/ rf#— 



+ jj^i* il + &c - + v ^ - V "H/ + te-^85) 



(P-Q/ + &C.)- (ty, ,-*,**„) (P„-Q/+ &c.) +t7,-t£ 

 + (*/>,- ?,»*,) (Qj-&c.) - (8p„- qj*„) (Q M - &c) +&c. 



since / a a: — 1 d x — — = / ax — — ; &c. = &c. 



t/ dx„ J dx„ <J x dx„ 



It appears, then, that the equation which we represented by 

 (j. 8 x t + it 8 x H = is 



5- fjdV 5, /»**, dV ,' /»*', dV 



8.T, / e?.r — boy// dx— h ox-. / dx — — + 



'c/ di, *"t/ di/. '%/ dx„ 



x„ ' x tl Jl x lt " 



^///^^+&c. + V / 8 l r / -V> // +(^ / -i 3 > / )(P / -Q/ + &c.) 



-$.y"-p,M,) (P„-Q/ +&a) + to - qJiKQ-ZLc) 



- to - & 8 O (Q.v- &c) + &c. = (3) 



If V contains none of the quantities x / y / x /t y tfl then since 



^ V r. dV A dV „ , dV n . . r 



r; — = - — = - — = and -: — = 0, the equation tor 



d x, dy t dx u dy lt 1 



determining the limits is 



V'M - V //K + ®y,-p,**.) ( p , - Q/ + &c) - 



to, -J^*,,) (P / ,-Q / / + &c) + {tp-q^x) (Q, - &C.) - 

 to, - ft 8 »J (Q.- &c) + &c = (4) 



By means of this equation, then, we can determine the limits 

 when they are not involved in V; but when they are involved 

 in V, we must use equation (3) ; and therefore in either case 

 we can determine the arbitrary constants as was required. 



We will now resume the consideration of the value of 

 W / — W //} which in consequence of the preceding results is 

 reduced to 



x ,l 



2Em / w"+Fw' 2 + &c.} + &c. 



Now, 



