Fundamental Equation in general 1 lieory oj Relativity . 287 



For, on putting these values of the (j>//u) in (6), and then 

 going back successively through (6), (5), (3), we get for the 

 right-hand member of (3), 



(?i 2 + it - U + HY- + (Sf.fs) 2 + (2f 2 f s ) 2 + (2f 4 f 3 ) 2 , 

 which is identically 



that is, ds 2 . Hence (3), and therefore (1) which is equiva- 

 lent to (3), is identically satisfied by the indicated values of 



■s, %U #2> #3> X i' 



4. The same device of making the solution depend ulti- 

 mately upon an identity connecting sums of squares, can 

 obviously be applied to find parametric solutions of other 

 equations occurring in general dynamics. It will be suffi- 

 cient to indicate the identity applicable to 



ds 2 = ^aijdXidxj (i,j=l, 2, . . ., re). . ■. (8) 



Suppose that the quadratic form on the right of (8) is 

 algebraically reduced, in the ordinary way, to a sum of 

 n squares, 



n 



ds 2 = ^(ljiid,v l + b i2 d.v.2+ . . . +b iH d l v n y, . . (9) 



such that the determinant |6^| does not vanish. As before, 

 special cases in which this reduction is impossible may arise ; 

 but they present no essential difficulty. Then, for n>l, we 

 resolve re— 1 in any way into a pair of factors r, s so that 

 ?i = rs+l ; and put 



A r =,2£ 2 3 B.=ii&», 



where f;, rjj are arbitrary functions of a parameter u. Then 

 the identity leading to a solution of (9), and hence of (8), is 



(A r + B S ) 2 = (A,.-B 5 ) 2 + 4A,B 5 . . . . (10) 



For, on multiplying out A?B S , the right of (10) is a sum of 

 rs + l = ?i squares, viz., 



(A r + B 8) 2 = (A r -B s ) 2 + k Z(2% iVj y ; . . (11) 



i=l ;'=1 



