ON THE RECENT PROGRESS OF ANALYSIS. 279 



To satisfy ourselves that these are the roots required, we 

 observe that </> (mw 4- nisi) = for all integral values of m and n. 



Hence the general form of the roots of our equation is ^ - ; 



o 



but it will be found that if we give any values not included in 

 the above table to m and ft, the resulting expression can be re- 

 duced to one or other of the forms we have specified in virtue of 

 the formula (0) =<{(- l) m+ " + may + nvri}. E. g. The non- 



,-,-1 ,-, , , 5co + 2m . - . , 4 (co + iifi) 



tabulated root < -- - -- is equal to our sixth root (j> - -- -, 



since the sum of their arguments is 3<w + 2m, and the sum of 

 3 and 2 is an odd number. 



On considering our table, we observe that it consists of 3 + 1 

 horizontal rows, each containing 3 1 terms, and that the argu- 

 ments of the terms in each row are connected by a simple 

 relation ; that of the second being double that of the first. If 

 we were to replace 3 by any odd number p, we should get an 

 equation of the p 2 degree, whose roots, setting aside zero, might 

 similarly be arranged in p + 1 rows, each of p 1 terms, the 

 arguments of the terms in each row being as 1, 2, 3, &c. 



Moreover, sin is rationally expressible in sin , and 



generally sin is so in sin - , n and p being any in- 



tegers we please. So too are all the terms in each horizontal 

 row of our table, whether for the particular case we have written 

 down, or for that of any odd number, rationally expressible in 

 the first term. 



Hence it may be shown that when the divisor 2n + l is a 

 prime number, an equation whose roots were the terms in any 

 horizontal row could be solved algebraically, by a method 

 essentially the same as that of Gauss, just as we can solve the 



o . 



equation the type of whose roots is sin - . But to construct 



J ^ 271 + 1 



this equation, i. e. to determine its coefficients, requires the so- 

 lution of an equation of the same degree as the number of 

 horizontal rows, i. e. of the degree 2n + 2. And this equation 

 is in general insoluble. The difficulty we here encounter may 

 be expressed in general language, by saying that although we 



