382 VIII. NEWTONIAN POTENTIAL FUNCTION. 



Let us now make an infinitesimal transformation of the curve 

 as in 30. Then the change in the integral is 



B 



38) dI== 



The last three terms can be integrated by parts, giving 



B B 



39) f E s ddq s = E s dq s - fiq s dE s , (* = 1,2,3), 



A A 



and, since the integrated terms vanish at the limits, 



40) dI=J(dE 1 dq + dE, dq 2 + dE^ dq z -dE 1 dq,- dE, dq 2 - dE 3 d &). 



Performing the operations denoted by d and d, as on p. 85, six of 

 the eighteen terms cancel, and there remain the terms, 



Now the changes dg 9 , dq%, d# 3 , dq s , in the coordinates correspond 

 to distances , , 



measured along the coordinate lines, and the determinant of these 

 distances, 



is equal to the area of the projection on the surface q of the 

 infinitesimal parallelogram swept over by the arc ds during the trans- 

 formation. Calling this area dS, and its normal n, we have 



dq s - dq B dq^) = cos 0%) dS. 



If we now continually repeat the transformation, until the curve 1 

 joining AB is transformed into the curve 2, the total change in 1 

 is equal to the surface integral over the intervening surface, 



42) 



