54 DEFINITE INTEGRALS. [INT. 111. 



integral will change, and we shall now seek an expression for the 

 variation. We have 



ds ds 

 dX ~ dX ~ dX ~ 



XT <> 



NOW & 



dx dy dz 



^dx d (Sx) 



and & -y = -A . 



ds ds 



We may perform upon the term 



A 



an integration by parts 



IB 



where XSx j signifies that from the value of the function X&K at 

 the point B we subtract its value at A. Now 



dX dXdx dXdy dXdz 

 i ^ _i 



ds dx ds dy ds dz ds ' 

 Performing similar operations on the other terms we have 



*T fr* L ro i. 7* \ IB - ft^X , .dX^ .dX\dx 

 o 1 = (A ox + i oy + Zbz) 



. 



ds \dx dy ' dz J ds 

 dx dX dy dX dz 



- _ 



fd x y z 



___ f\fV I _ _ T I _ _ L j -, _ _ 



V dx ds dy ds dz ds 



fdYdx dYdy dTdz\ 



y\*" j" + ^~ :j+^ r ) 

 9 \dx ds dy ds dz dsJ 



(dZ_ clx 3Z dy dZ dz\] , 

 *(fads~ + tyds + dz ds)\ 



Now if in the variation the ends of the curve A and B are 

 fixed, &, &/, Sz vanish for A and B, and the integrated part 



