ON THE RECENT PROGRESS OF ANALYSIS. 61 
On considering our table, we observe that it consists of 3 + 1 horizontal 
‘rows, each containing 3 — 1 terms, and that the arguments 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* degree, whose roots, setting aside zero, might simi- 
larly 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 rarionally expressible in sin = and generally sin abn 
7” and p being any integers we please. So too are all 
, eas on 
is sO in sin 
Qn 
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 2 + 1 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 equation the type of whose roots is sin se = I" But to con- 
t 
_ struct this equation, z. e. to determine its coefficients, requires the solution of 
an equation of the same degree as the number of horizontal rows, 7. e. of the 
_ degree 2+ 2. And this equation is in general insoluble. The difficulty 
_ we here encounter may be expressed in general language, by saying that 
_ although we can pass from one root to another along each horizontal row, , 
__ yet we cannot pass from row to row. 
Our table, however, has the remarkable property, that supposing, as we 
may always do, 2% +1 to be a prime number, all the roots are rationally 
expressible in terms of any two not lying in the same row. This depends on 
a property of the function ¢, which it is very easy to demonstrate, and it is 
intimately connected with the relations which exist among the terms of the 
same row. 
If, then, which is the case for an infinite variety of values of the modulus, 
We can express any root rationally in terms of another of a different row, 
say in @ cis ae 
2n+1 Qn+1 
_ it appears that not only are the roots all expressible in one, but they are so 
_ in such a manner that the functional dependencies among them fulfil a cer- 
_ tain simple condition, which, as Abel shows in a separate memoir (Crelle, iv. 
_ p- 131; or Abel’s works, i. p. 114), renders every equation, all whose roots 
_ are rationally expressible in terms of one, algebraically soluble. ~ 
_ To take the simplest case, the are of the lemniscate may be represented by 
, all the roots become rational in terms of ¢ - Moreover, 
dz death 
_ the integral Fag If ¢ be the function inverse to this integral, we have 
| the simple relation between roots of different rows, ¢ Poa i= t@ wat ? 
_ w being in this case equal to w. 
__ To apply what has been said to the solution of the general equation for 
determining ¢ a in terms of 9 (2% ain 1) a, it is sufficient to remark that the 
anscendents introduced in considering the relations among the roots of this 
7 k 2 Qar : 
a ee uation, are simply @ ree and @ ry or at least may be algebraically 
expressed in terms of these two quantities. 
___ The remainder of the first memoir contains developments of the functions 
