63] HAMILTON'S PRINCIPLE. 123 



Then the coefficients of every dx s vanish, and since we may 

 take the dy's and dz's arbitrarily, in order for the sum to vanish 

 we must have for every dy s and dz s , 



dG_ dF_dG 



ty.~ '*" dz.-**.' 



The function -G is called the reciprocal function to the 

 function F with respect to the variables x^ ...tc n , fc> r we have 

 the reciprocal relations 



or: 



Two reciprocal functions have the property that the partial 

 derivative of either with respect to any variable of reciprocation 

 contained in it is equal to the corresponding variable replacing the 

 original in the other function, whereas the partial derivative of one 

 function with respect to any variable not of reciprocation is the 

 negative of the derivative of the other function with respect to the 

 same variable. 



In case the function F is homogeneous of degree /c in the 

 variables of reciprocation 



*^i j *^2 > ^n 



the theorem becomes more striking, for then, by Euler's theorem 

 dF dF dF 



and the reciprocal function is simply a multiple of the original 

 function. 



If the original function is of degree two, the reciprocal function 

 is identically equal to the original function. We have thus a 

 striking example of the remark made at the beginning of this 

 section, for here the derivative of the function when expressed 

 in one form by a variable z is exactly the negative of the derivative 

 by the same variable of the function expressed in the other form. 

 In this form the theorem will be frequently used hereafter. By 

 means of it the equations of motion may be transformed from 

 Lagrange's form to that given them by Hamilton. 



