578 NOTES. 



the /'s and g'a being real linear forms in the #'s, any of which, but 

 not all, may be zero. Separating the real and the imaginary terms. 

 we obtain 



r==n \ r = n 1 



30) 9 + ^ 



From the two forms 



the whole sheaf may be obtained. Solving 30) for g> and 



32) A cp + y 



which we may write 



r= ^~ 1 / ! v 



33) I Cp -f ib = >Yy r ^#r) [9r H #r) 



^_ \ f* ./ 



r=l 



where 



^ ^ 



a quadratic in ft, giving for every real value of A, a real value of jii. 

 Now each of the forms 33) vanishes for values of # 15 . . . x n other than 

 zero, which satisfy the n 1 linear equations 



and is accordingly indefinite. Conversely if there is in the sheaf a single 

 definite form, the roots of /"(A) = are all real. Now in the mechanical 

 application, the form go, which is proportional to that given by the value 

 A = oo, is the kinetic energy, a definite positive form, consequently the 

 reality of the roots is proved. 



If A x is one of the real roots of the equation /"(A) = the form 

 A*9> + ty being singular, can be expressed in terms of less than n linear 

 functions of a^, ...#, say y^...y n \. Let y n be any other linear 

 function of the #'s, such that the determinant of the functions / 1? . . . y n 

 is not zero, then we can express the function <p in terms of these n 

 variables /, and if it is definite and positive, the coefficient of every 

 square will have a positive sign, accordingly, as in 12), 13), we may 

 separate off from (p a square # x 2 , where z contains y ni and accordingly 

 we have 



n Q) ~\~ ty == (A A.;/ ) OP ~p AX (p ~T~ ty 



35) -a n 2 + 1 ' 4- f 



where A 99'+ i// contains only the n 1 variables y 1? . . .y n \. Now a 



definite form in w variables remains a definite form in n 1 variables 



if we put any variable equal to zero, consequently g/ is, like g>, a 



definite positive form. 



