212 SCIENCE PROGRESS 



Thus in the equation 



2X 1 + 1 1 x* — io#* — 26# 4 + 31*' + 7 2 x* — 230* — 348 = o 



the number 4, which is equal to the greatest of the quantities 



.10 .26 . 230 . 348 



1 -J * 1 4 • 1 4 — ' 1 4- • 



+ 13 ' + 13 ' ^ 116 ' I+ 116' 



is a superior limit required, and if we change the signs of the 



alternate terms, we shall have 1 -\ or 7, a superior limit 



2 



of the roots of the resulting equations. Therefore the real 



roots of the first equation lie between 4 and — 7. 



We have in quite recent times many papers on the subject 



by Landau in 1907, Fejur in the same year, Carmichael and 



Mason in 19 14. These two last mentioned give the interesting 



result that — 



All the roots of the equation 



a 



x n + «i# n_1 + a 2 x n ~ 2 + . . . + a n = o 

 are in absolute value less than or equal to 



J I + I #1 I * + I «2 I 8 + • • • + I On T 



It was natural, that after the complete solution of the 

 cubic and biquadratic equations, especial attention should be 

 given to the quintic, and this does not escape Dary's notice, 

 and he is cute enough to choose the Bring-Jerrard form for 

 his example. 



It is only since Abel's time we have been familiar with the 

 fact that the quintic could not be completely solved without 

 the aid of elliptic and hyper-elliptic transcendentals, and we 

 now know that the complete solution of an equation of the 

 fifth and higher degrees, by means of the elementary operations 

 of addition, subtraction, multiplication, division, and a finite 

 number of root extractions, can only be obtained when the 

 group of the equation is soluble. 



Modern writers on the subject such as Lemeray, Isenkrahe, 

 Ross, have occupied themselves with considerations of the 

 various forms of approach to the roots, and have also con- 

 sidered the rate of convergence. 



Dary, who lived 200 years before the rigid ideas of con- 

 vergence had taken hold, was not likely to have realised the 

 difficulties involved, which were such as to make De Morgan 



