374 Mr. C. E. Van Orstrand on 



The preceding expressions are not always sufficiently conver- 

 gent* to be useful in practice. They may be made convergent, 

 however, by substituting an approximate value o£ x in the 

 original equation. It thus becomes possible to obtain all of 

 the real roots of any polynomial. After one or more of the 

 roots a x , a 2 , a 3 , ... have been obtained, use may be made of 

 the relations f 



« 1= (ai« 2 + «i«3+ ... a»_i«») 



a 2 — — («l«2 a 3 + a l a 3 a 4 + • • • a»-2«»-i««) 



a n ( — l) n («i«2 a 3 ••• *»-i*»)> 

 in the evaluation of the remaining roots. A method for 

 determining all of the real and imaginary roots from a single 

 series has been given by McClintockJ. 

 Following are some examples : 



X 2 X 4 X 6 

 (1) y=coshX-l=2J +£f + (n+ — 



X X 2 X 3 



X 2 = X=- - 2 2-|-i -3 1 -4 ■ 1 ~5_ 



"' 12 ^90* 560' ^3150 "" 

 o 1 o 4 ., 1 . 16 , 



2 f 1 ^3 jji ,.7 



(2) ^^Jo ^ d — '-TT3 + 2T5 - 5T7 + - 



v/tT 1 . ■ 1 z 1 



*=*+* i. 1.5 ,127 ^ 4369 34807 , 



:t- 3 s -t- 30 ~ ^630* + 22680 + 178200" + '" 

 = 0*8862 2693t/ +0-2320 1367/+0-1275 5618/ 

 + 0-0865 5213/ + 0-0649 5962/ + 0*0517 3128/ 1 + ..... 



* For a proof that the reverse series converges in the same domain as 

 the original series, see Harkness and Morley's ' Treatise on the Theory 

 of Functions/ p. 116. 



I Burnside and Panton's « Theory of Equations/ Chapter III. 



% Bulletin Am. Math. Soc. i. 1894, p. 3; Am. Jour, of Math. xvii. 

 1895, pp. 89-110. Merriman and Woodward's ' Higher Mathematics/ 



