29] VARIATION OF INTEGRAL. 81 



not vanish. We must as before introduce the independent variable, 

 writing 



d j (pdx = d I yxdt = I (dtp x' + <pdx')dt 



(dtp dx -f g>ddx). 



It may, on occasion, be more convenient to use these more general 

 formulae, not supposing the variation of any variable to vanish.) 



If the limits are varied, we have, indicating the part of the 

 change in I due to the change in either limit by a suffix, 



*? * r h f h 



\I = I cpdt lcpdt= lcpdt = 

 J J J 



which are to added to the part already found. 



In the application of the calculus of variations, we often 

 encounter problems involving a number of independent variables, so 

 that we deal with partial derivatives, and multiple integrals. The 

 principles here given will however suffice for the treatment of all 

 the usual questions. 



As a celebrated mechanical example of the use of the Calculus 

 of Variations let us consider the question: What is that curve along 

 which a particle must be constrained to descend under the influence 

 of gravity in order to pass from one point to another in the least 

 possible time? 



Since v = ^ S .> we have for the time of descent t = I - > or 

 dt 



making use of the equation of energy 27, 27), 



= I - 

 J v 



Let us take for the independent variable corresponding to t above 

 the vertical coordinate 0. We suppose the motion to take place 

 in a vertical plane. We have then 



If now we make an arbitrary infinitesimal variation of the 

 curve, if t is to be a minimum we mast have the term of the first 

 order in s vanish, 



dt = 0. 



WEBSTER, Dynamics. 6 



