148 



Lord Rayleigh on the Maintenance of 



where n is any positive or negative integer or zero. And 

 these are all the roots the equation admits of ; for each of the 

 expressions denoted by [w] is of two dimensions in c, and may 

 be regarded as introducing into the equation the two roots 

 2n + c and 2n — c Q . Consequently the roots are either all 

 real or all imaginary; and it is impossible that the equation 

 should have any equal root unless all the roots are integral." 



On these grounds Mr. Hill concludes that S(c) must be 

 such that 



®(c)=A [COS (7TC)— COS (7TC )] . . . (11) 



identically, where A is some constant independent of c ; whence 

 on putting c = 0, 



^(0)=A[l-cos(^ o )], .... (12) 



in which, if we please, c may be replaced by c. The value 

 of A may now be determined by comparison with the parti- 

 cular case ©! = (), © 2 = 0, &c, for which of course c= v / ©o- 

 Thus if S\0) denote the special form then assumed, i. e. the 

 simple product of the diagonal constituents, 



® / (0)=A[1-cos(tt n /©o)], • (13) 



and 



l — cos (7rc) sin 2 (^7TC) _2)(0) . . 



l-cos(7TA/e^ ~~ sin 2 (i*V© ) ~~ WW)' ' [ } 



The fraction S(0)-^S) / (0) is denoted by 0(0). It is the 

 determinant formed from the original one by dividing each 

 row by the constituent in the diagonal, so as to reduce all 

 the diagonal constituents to unity, and bv making c vanish. 

 Thus 



-cosM J n y _ _ (15) 



where 



0(0) 



1 — cos(wV@ ) 



©! 



©< 



a 



©4 



•* + * 4 2 -© 4 2 -© 4 2 -© 4 2 -© 



©i 



2 2 -© 



©2 



+ 



©1 



©, 



©, 



2 2 -© 2 2 -© 2 2 -© 



0, 



2 -© 2 -© 



©3 ©2 



+ 1 ~ 



©1 , 



@, 



a 



2 -© 2 -© 



©1 



2 2 -© n 2 2 -©,, 2 2 -© r 



2 2 -©. 



©, 



0, 



©, 



a 



" 4 2 -© 4 2 -© 4 2 -© 4 2 -© 



(16) 



