On a Theorem relating to Equations of the Fifth Order. 257 
If the tube, completely filled with liquid, be raised as repre- 
sented in fig. 4, it will be found that the liquid will stand in the 
tube some distance higher than the level of the liquid in the 
vessel. The column of elevation C D, measuring the capillary 
action, was found, under the same circumstances, to be nearly the 
same as in the preceding cases. Here the capillary action is 
restricted to the orifice of the thin plate, the diameter of the 
tube EB with which the experiment was performed being about 
oneinch. This experiment strikingly shows that the height of 
the column measuring capillary action for any given liquid at a 
constant temperature depends (chiefly, if not entirely) upon the dia- 
meter of that portion of the tube immediately in contact with the 
upper surface of the liquid. 
March 18, 1861. 
XLI. On a Theorem of Abel’s relating to Equations of the 
Fifth Order. By A. Cayiuny, Esq.* 
P / SHE following is given (Abel, Giwvres, vol. x1. p. 253) as an - 
extract of a letter to M. Crelle :— 
“Si une équation du cinquiéme degré, dont les coefficients 
sont des nombres rationnels, est résoluble algébriquement, on peut 
donner aux racines la forme suivante, 
a=c-+ Aata,? do5ag> + A,a,3 dy3 458 a5 + Agaos ag 
ou 
as 
ay? + Asay a? 5 a3 ag ; 
a=m+n/1+2 4VAl te +e? 4 -V1 +e), 
a,=m—n/1+e Perv h(l +e —V1+¢ +e), 
dg=m+n/1Fe—V Al +e+ V1 +e), 
az=m—n/1LEE—V h(l+e— V1+e +), 
A =K+4+Kla +K"a.+K"aa,, A,=K+K!a,+K"a,+ Kaya, 
A,=K+K'a,+K"a + K"aa,, Az=K+Kla,+ K"a,+ K"aja,. 
Ties quantités c,h, e, m,n, K, K!, K", K'" sont des nombres 
rationnels. 
“Mais de cette maniére léquation z°+axr+b=0 n’est pas 
résoluble tant que a et 6 sont des quantités quelconques. J’ai 
trouvé de pareils théorémes pour les équations du 7*™¢, 11*™, 
13°m°, &c. degré. Fribourg, le 14 Mars, 1826.” 
The theorem is referred to by M. Kronecker (Berl. Monatsb. 
June 20, 1853), but nowhere else that I am aware of. 
It is to be noticed that in the expressions for a, a, dq, a3, the 
| * Communicated by the Author. 
Phil. Mag, 8. 4, Vol. 21. No. 140, April 1861. S 
