OF DEFLECTIVE FORCES. 119 



" differentiation from curve to curve." For a further 

 illustration of this method, see article 289.] 



266. Theorem. The path of a moveable 

 body is always such, that the fluent fods^ 

 taken between its extreme points, is less than 

 in any other curve subjected to the same 

 conditions. 



Since v^zzc + 2f(Pdx-{-Qdy + Rdz) (264, Cor. 1) and 



since the coefficients of the variation S and of the fluxion 



d are the same, it follows that v^v= F^x -\- Q^y + R^z ; 



dx 

 consequently in the equation (f, 264) 0=oa: (d-r- — 



Pdt) + hj{d^-'Qdt) 4- Sz(dT^--jRdO, we may substi- 



stitute vMtfor P^xdt+ Q^ydt-hR^zdt, and it will become 



z= Sard ^+ M-^+ ^zd^-^v^vdUih); now if the fluxion 



of the curve be called d*, we shall have ds=vdt, and d^* 

 = dx2 +d?/2+(iz2, consequently v8vdfi=d5§i; ; and, taking 

 the variation of d*^, diSd* = dxSdx + dy^dy 4- dzMz 

 = vd^Sd*. Now d {dxdx) = dxdSx + dxd^x ; but since 

 d^x = ^dx (265), dxgdx = dxdgx z= d (dx^x) - ^xd^x, 

 and the same substitutions being applied to the coordinates 

 y and z, the equation for ds^ds is transformed into vSd5= 

 d(dx^x-\-dy^y-\-dz^z) _ , dx . , dy ^ ,dz 



IT ^^^d7-^2^^df-^"^d7^ ^*^^^^' 



by means of the equation (hj becomes v^ds = d 



dx^x + dyh/ + dz^z ^ , ^ , , ^ ^ , 

 -^Tf^ v^vdt: hut v^vdtzzds^v ; and ^(vds) 



5»j . J ^ ■,dx^x-\-dy^y-{-dz^z , ^|. ^. 



=iv6ds + dsdv = d ~ : and taking the 



flirent on both sides with respect to d, 8/rd» = c -j- 



