94 G. P. YOUNG ON ABEL'S FORMS OF THE ROOTS 
where p, g, and e are rational. Also let Q, be a rational function of 4; Q, the same 
same rational function of #, ; Q, the same rational function of #,; and Q, the same rational 
function of #. Then Abel’s form for the root of the solvable quintic wanting the second 
term is 
1 
5 
1 1 nl 
Q (A, 6, 8, &)" +0 (aa a A)” + Q, (68 a 8) + Q, (8, 6, A, 8)". (1) 
Can the value of x in (5) be thrown into this form? It is the object of the paper to show 
that it can. 
II.— PRELIMINARY EXPLANATIONS. 
§ 2. If s be put for Ts the four values of À, obtained from equation (3), are 
A=S[ItVA+SH+VI2A+S+2V(+8}] 
A= [1-vd+s4+vj{204+s)—-2vd+s)}] 
W=rlit+vdt+s)—v {20+ s)+24+0+8)t] 
À -[1—-vd+s)—vj204+s8)—2v(+s){]. 
| 
2 
4 
3 
| 
These expressions, À, À, À, À, circulate. That is to say, the changes that cause A, to 
become A,, cause À, to become A,, and A, to become A, and A, to become A,. For, in order 
that À, may become 1,, we must alter the sign of 4 (1+ 5°), and take the new radical thus 
produced, viz.  $2 (1+s) —2 4 (1+5)$, with the positive sign. To make the same 
changes on A,, we must first express the radical / {2 (1+ s°) —2 ¥ (1+5)}, which does 
not occur in that form in À,, in terms of the radicals present in À, In fact, 
. ee 2s/(1+s’) 
2s/(1+s) ; 
2il+s)+2vd+s')} 
By making the same changes on A, that were made on À, s A, becomes 

Therefore sÀ,—1— 4 (1+ 5) + 
2s/ (1+s) 
20 +s8)—2yY(1+8)}’ 
or lt+Vd+e)—vj2Zi+s)te2ya+syt, 
which is the value of sA,. Thus, A, has become 1, In like manner, À, becomes 1;, and 
À, becomes A.. 
§ 3. Since 4 is, by (4), a rational function of À, let 6,, 4, 4%, %, be the values of 4 corre- 
sponding respectively to A,, 4,, 44, A4 Then the terms 4, 4, 6, 4, must circulate with 
AA AA PACS Osis 

1+YQ04+8) — 
( A 
Om — — À! 0! a, 
Q is, by (4), a rational function of A. Hence, if the four values of Q, corresponding respec- 
tively to A,, A. Ay As, be @, Q, Q, Q, these expressions, Q,, Q, Q, Q, circulate with 
nains 
