190 BOOK OF MATHEMATICAL PROBLEMS. 



be equal to the sum of the homogeneous products of r dimensions 

 of the n quantities a t , a,, ... a m . Prove also that 



jg^+ug^+^+jgjg 



' H j t /... + "-i i _(i + .... + iy. 



(/ d \ Cb CL j 



900. The equation 



x + 

 will be an identical equation if 



" 1 = 0. 



901. If n be a prime number, and a an impossible root of 

 the equation #"=1, then will 



1 a a' a 8 a"" 1 n 



+ + - m + = + ... + 



m+l ra + a w + a* ra + a 8 w 4- a" ' 1 (- /?*)" 



902. If the system of equations of which the type is 



be true for integral values of r from r=l to r = w 4- 1, they will 

 be true for any value of r. 



903. Having given 



cos na 4- p l cos (n 1) a + p a cos (n 2) a + ... + p n = 0, 

 sin na+p v sin (w 1) a + p a sin (n 2) a + ... + jo, = ; 



prove that 



1 +p l cos a +p g cos 2a -f- ... 4-^ cos na= 0, 



and 1 4- p l sin a + p t sin 2a 4- ... 4- p n sin Tia = 0. 



904. Find the commensurable roots of the equations 



(2) 6* 4 -.a'- 2*' -27* + 18 = 0.. 



