Vibrations by Forces of Double Frequency. 147 



It will be convenient to give here a sketch of Mr. Hill's 

 method and results. Remarking that when © x , © 2 , &c. vanish, 

 the solution of (3) is 



w = Ke ict + K / e- ict , (5) 



where K, K7 are arbitrary constants, and c = s/ (© ) ? ne shows 

 that in the general case we may assume as a particular solution 



= t n b ni 



ict+2int 



(6) 



the value of c being modified by the operation of © 1? &c, and 

 the original term b e ict being accompanied by subordinate 

 terms corresponding to the positive and negative integral 

 values of n. 



The multiplication by ©, as given in (4), does not alter the 

 form of (6) ; and the result of the substitution in the differ- 

 ential equation (3) may be written 



( C + 2m)^-2»© ro _A=0, .... (7) 



which holds for all integral values of m, positive and negative. 

 These conditions determine the ratios of all the coefficients 

 b n to one of them, e. g., b , which may then be regarded as 

 the arbitrary constant. They also determine c, the main sub- 

 ject of quest. Mr. Hill writes 



[n]=(c + 2») 2 -®„; (8) 



so that the equations take the form 



,+ [-2]5_ 2 - ©! b^-S 2 b - © 3 ^-© 4 Z> 2 - 



- @! Z>_ 2 + [-l]^-©! b Q - © 2 h-© 3 b 2 - 



- © 2 b- 2 - ©! b^+[0]b ~ @! ^-©2^- 



- © 3 &-2- ©2 ^l-©l5o+[-l>l-©l^- 



- ©4 &-.- © 3 6-i-G. V- ©i 6i+[2]6a- 



..=0, 

 .. = 0, 



.. = 0, 

 .. = 0, 

 .. = 0, 



The determinant formed by eliminating the 5's from these 

 equations is denoted by S(c); so that the equation from which 

 c is to be found is 



©(e) =0 (10) 



The infinite series of values of c determined by (10) cannot 

 give independent solutions of (3), — a differential equation 

 of the second order only. It is evident, in fact, that the 

 system of equations by which c is determined is not altered if 

 we replace c hy c + 2v, where v is any positive or negative 

 integer. Neither is any change incurred by the substitution 

 of — c for c. "It follows that if (10) is satisfied by a root 

 c=c , it will also have, as roots, all the quantities contained 

 in the expression ±c + 2n, 



L2 



(9) 



