84 in. GENERAL PRINCIPLES. WORK AND ENERGY. 



The curve described by a point on the circumference of the 

 rolling circle is called a cycloid. Obviously from the symmetry of 

 the case the curve is symmetrical about a vertical axis on which the 

 point lies when # = 0. Thus our equations of the .brachistochrone 

 show that it is a cycloid with vertical axis. 



The arbitrary constants of integration, 6 and d, are determined 

 by the two points through which the curve is to pass. The discovery 

 that the cycloid is the brachistochrone for gravity is due to Jean 

 Bernoulli. 



30. Dependence of Line Integral on Path. Stokes's 

 Theorem. Curl. Consider now our line -integral, 



B 



I=J(Xdx + Ydy + Zds). 



We have first to introduce an independent variable corresponding to 

 the t of the previous section, variation of which shall cause the point 

 of integration to move along a given curve. Let us call this s, 

 which to fix the ideas may (through this is unnecessary) be considered 

 as the distance measured along the curve from the point A. Thus 

 we write 



The functions X, Y, Z, being given for every point x, y, 0, the 

 integral I will in general depend on the form of the curve AS. If 

 we make an infinitesimal transformation of the curve, the integral 

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

 We have 



Now 

 and 



~dx _ d(8x) 

 ds ds 



We may perform upon the term 



B 



r d(dx) 



