134 A METHOD OF SOLVING THE EQUATION ~+Qw = Q. [6 



the angles involved being 



k-lt + P, k-Gt+fi, k-8t + /3, 



Proceed in this way as far as may be necessary for the degree of 

 approximation desired. Then the result of substituting 



W W + St'W + 8,1V + ... 



in the equation 



will be 



... cos t + + _ l + l _ l + ,_ l + ... cos - 



where the coefficients cos(^ + /8), cos (k 2 1 + /8) involve k and c_^. If these 

 coefficients be equated to zero \ve have means of determining k and c_,. 



[30 May 1884.] 



[To illustrate this method somewhat further ; substitute 



tu = coskt + c_ l cos (k 2) t 



in the differential equation ; then we get 



then take the equation 



72$^ 



-r + q& w + Q-* cos (k - 4) * + Q, cos (k + 2) t, 



whence 



cos ~ 4 f + cos 



( k - 4y - 



substitute this in the expression 



(2q, cos 2t + 2q, cos 4 + . . . ) 8^, 

 and we get 



