524 Sir William Thomson on Stationary 



or > • \*°) ; 



W=(l + J)(l + J)(l + g) & J 



where ft 2 , ft 2 , &c. are real positive numerics, while ft 2 is 

 real positive or real negative according as b is greater than D 

 or less than D. 



Resolving now the reciprocal of W into partial fractions, 

 we find 



w-T~ Z 77" ? + r *i + ' ' ( ); 



+ X 2 + ft 2 ft 2 



where 



-1 -2 (l-D/6)cosft 



N, 





(dW\ " D lb - coa 2 6i 



(30) 



(l-D/&)sinft 

 ft(l-&/D.cos 2 ft) 

 Fori=l and D>5, ft is, as we have seen, imaginary (its 

 square real negative), and for this case the formula (30) may- 

 be conveniently written 



jP/ft-lH^ + e-i) 



and the equation for finding <r x is 



€ t 1 + <:-*,_ _ ( 6 ^i- e-^i )=0 .... (32), 



an equation which has one, and only one, real root when 

 ~D>b, and no real root when D < b. 



When b/~D is given, it is easy to find, as the case may be, 

 o"! of (32) or ft the first-quadrant root of (27), by arithmetical 

 trial and error ; and the successive roots ft, ft, &c. more and 

 more easily, by the solution of (27). It is to be remarked 

 that, whatever be the value of b/D, these roots approach more 

 and more nearly to the superior limits of the quadrants in 

 which they lie : thus if we put 



«•=* (<-«»-* (33), 



■we have 



^=(-D-^Sf[(^— ] ■ 



" l lj l-&/Dsin 2 « t - •••••• W* 



