132 MR. JAMES COCKLE; RESEARCHES 
then, if the given equation be of the form 
+ —5Qu'4-H=0 . 0... sheen eee (1), 
there subsists the relation 
7S Vea | i a eel (2). 
§ 4. 
When one of the six values of @ vanishes, (1) admits of 
finite algebraic solution. 
It is unnecessary to repeat the investigations in the 
course of which I arrived at this criterion of solvibility, 
or to exhibit the complicated formula by which, in the 
general case, it is expressed.* But the solution, if any 
exist, of the general equation of the fifth degree, will con- 
tain a term of the form 7”@, where © vanishes with @. 
§ 5. 
When (1) holds, the conditions of homogeneity indicate 
65+ aQh& + BQ‘ + yQVE’# + (SQ°E + eH’) 0 
22 GA iQ * ERO) 0 i A 8 38. A (3) 
as the form of the equation in @. And in determining 
a, 8, &e., we may make 7=0, for E* does not enter into 
the criterion alluded to. 
§ 6. 
Let y(w—h) or V be the value of U when each of the 
roots of (1) is diminished by er Then 
V=U-DU.- h+DU +5 57 DU: Tt = gt he. 
where D is the differential ae = 4 tate 
But 
DU =Se'w4+ 25vwe =- 5Q, D?U=450?4 103vw=0, 
DU =4830=0, D*U = 240, 
consequently 
WW SORA 10h Wk > femee e (4). 
* See Phil. Mag., May 1858, p. 389. 
