NOTES, 



NOTE I. 



DIFFERENTIAL EQUATIONS. 



The differential equations of mechanics are of the type known as 

 ordinary, as opposed to partial, that is they involve a number of functions 

 of a single variable, the time, and the derivatives of these functions with 

 respect to that variable. Suppose for simplicity that we have three func- 

 tions #, /, of the variable t y and that instead of being given explicitly, 

 they are defined by the equations 



If we now differentiate these equations, bearing in mind that #, /, are 

 dependent on t, we obtain 



dF, dx dF, dy dF, dz dF, _ 

 dt " 1 " dy dt "*" ds dt 1' dt ~ 



dF, dx dj\ dy dF, dz dj\ _ 

 3x dt " t " dy dt " l " dz dt "*" dt ~ ' 



dj\d^ dj\dy_ dj\d^ dj\ = 

 dx dt "" dy dt ~* dz dt ^ dt 



Suppose now that the functions F contained, besides the variables indi- 

 cated, certain constants, c 1? C 2 . . . Each time that we obtain an equation 

 by differentiation, we may utilize it in order to eliminate from the 

 equations 1) one of the constants c. Thus we obtain (since the partial 

 derivatives are given functions of x, y, z, f), instead of the equations 1), 

 the following, 



3) 



which, since they contain the derivatives -^i -j^i -^t are differential equa- 



tions, of which equations l) are said to be integrals. 

 If we again differentiate equations 2), we obtain 



, dF, d*x d*F, /day d*Ft dxdy _ 



~fa ~w + ~d^ (~dt) + 2 --^ 



