288 Fundamental Equation in general Theory of Relativity* 



and hence on putting 



s=§(A r + B s )du, 



hi^ + b i2 x 2 + . * . + bi n ®n =J </>; (w) du (i = 1, 2, . . ., n), 



. . . (12) 



where the functions <pi(u) are identical in some order with 

 the 2^i7]j, (Ar—Bg) in (11), we get, on solving (12) for the 

 %i, a solution of (9) and hence of (8). 



5. The cases of (8) in which the quadratic differeirtial form 

 on the right is reducible to a sum of 4/c + 3 squares, where 

 «=0, 1, 2, 3, . . . , are in many respects remarkable. In 

 those cases it is always possible, in several essentially 

 distinct ways, to obtain types of: general parametric solu- 

 tions in a form free from all quadratures. The same 

 property holds also when the right of (8) is reducible to a 

 sum of two squares. When the number of squares in the 

 reduced form is three, the most immediate interpretations 

 of (8) being to the theory of curves in three dimensional 

 space, whether Euclidean or not, the solution has a particular 

 interest. The cases of reductions to 2, 3, or 15 squares 

 present in addition many properties not shared by other 

 forms. The dynamical interpretation of the general case is 

 evident in terms of the generalized velocities, momenta, and 

 kinetic energy of a system, the last either in the Lagrangian 

 or Hamiltonian form. Hence it may be expected that if the 

 generalized coordinates of a system are 4/e-f3 in number, 

 the system will have special dynamical properties. In 

 addition to these quadratic differential forms, there are 

 many other classes of forms of degrees higher than the 

 second possessing a like property that the solutions of 

 equations between several such forms of the same kind may 

 be very readily obtained in parametric form, not, however, 

 free from quadratures. An account of all the cases men- 

 tioned in this note will shortly be published elsewhere. It 

 may be of interest to remark that all of the general solutions 

 free from quadratures, and those relating to forms of degree 

 higher than the second, first presented themselves in some 

 work relating to the theory of numbers. 



University of Washington, 

 Seattle, Washington, U.S.A. 



