350 GENEBALIZED COORDINATES 



Algebraic proofs of this theorem will be found in treatises on analysis, 

 or in Salmon's Higher Algebra, Lesson VI. The theorem will be readily 

 understood on considering a geometrical interpretation in the case in which 

 the number of variables is three. Calling the variables x, y, and z, the 

 equations 



/(*,y,*) = l, F(z,y,z) = l (191) 



will be the equations of concentric quadrics ; and since F is positive for all 

 values of x, y, z, the second quadric will be an ellipsoid. It is known that 

 two concentric quadrics, of which one is an ellipsoid, always have one real 

 set of mutually conjugate diameters in common. A transformation of the 

 type expressed by equation (194) enables us to transform to these axes as 

 axes of coordinates, and the equations of the quadrics are then of the 

 required forms 



i{i + s6+a 8 g = l, fcg + &g + /3 8 6 = l. ( 192 ) 



[Simple reasoning will show the truth of the geometrical theorem that 

 an ellipsoid and a second quadric always have one common set of real 

 mutually conjugate diameters. For a real linear transformation will trans- 

 form the ellipsoid into a sphere, and the second quadric into a new, but 

 still real, quadric. The principal axes of this real quadric are now real 

 mutually conjugate diameters for the sphere and the quadric, and on trans- 

 forming back, real mutually conjugate diameters remain real mutually 

 conjugate diameters.] 



The algebraic proof that equations (191) could be transformed into 

 equations (192) would, however, clearly not be limited to the case of three 

 variables, so that the theorem must be true for any number of variables. 



281. This theorem proves that we can find new coordinates 

 ^i> ^2* * ' * > ^n connected with ^, < 2 , -,< by relations of the 

 type 



*i = M"i + "12^2 + ' - + *!>, (194) 



such that, expressed in terms of these coordinates, the potential 

 and kinetic energies assume the forms 



W- W + 



T = P +& + ...+ /8.J. (196) 



The coordinates ty v ^ 2 , , ^r n are called the principal coordi- 

 nates of the system, or, by some writers, the normal coordinates. 



