84 Mr. J. Walker. Relative Retardation between Components 



Now the roots of equation (ii), p, q, r, s will be functions of sin i/v, 

 and expanding these in series proceeding according to powers of this 

 quantity, we may write generally 



p = 



2 = 



But 



, sin i sin 2 * 7 sin i . sin 3 i 



2p = 2c?! - , 2pg = C + c 2 , 2pgr = 2& 1 -^-+2& 8 -, 



sin 2 1 sin 4 i 

 ^pqrs = a n i-a<i + a 4 , 



and therefore equating the coefficients of like powers of sin ijv on the 

 two sides of these equations, the coefficients p, 5, r, 5 may in general 

 be determined in succession by means of linear equations, provided 

 we can determine _p , q , r , s 0t which may be at once done, since they 

 are the roots of the equation 



Moreover, when this method is practicable, the roots of the 

 biquadratic take the following form 



where 



s 



" l! 



sin- * sin * 



Ii " 



^ sin* ^ 



si n ^ sin * sin 



sn sn sn 



v 3 i> 5 



For suppose that this is true as far as the terms involving 

 ""*, and let a, /3, 7, denote the sum of the terms in w lt 7r 2 , 

 /> 2 respectively of an order less than n, then we may write 



q = a + 7 + /t2, r = y3 -j- 



