THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 193 
Riad h=07%, as. fe) 
e-5Q7 +E=0.... (2) 
g@=—5ee +H=0» .... We) 
# —5Ma*+EH=0°. . .-(f) 
(See Mr. Jerrard’s “‘ Mathematical Researches ;” Sir W. 
R. Hamilton’s “Inquiry into the Validity of Mr. Jerrard’s 
Method,” published in the Sixth Report of the British 
Association for the Advancement of Science; and M. 
Serret’s “Cours d’Algébre Supérieure,” Note V.) 
For (a), (@) and (ec), we have 
7 =—-7=3-5R-U, or U=77+3-5R . (a); 
and the corresponding relation for (d), (e) and (f) is 
=-P=-U, or U=" . . ~(a). 
Again: — For (a), we have 
G5 Ui PPE Se bP Bet ye ets (8); 
the corresponding relation for (4) and (c) is 
64-5 = 3 FR valine ot) (BS 
for (d), the relation is 
6+5U=0 Sh gle (BE) 
and for (e) and (/), 
2-bAU SRP appre droen(Be) 
and G2 Eau = pOME SI oe SI. i) 
respectively. 
It hence appears that (d) will be the most convenient 
form with which to deal in calculating the coefficients of 
the equation in @. 
27. We know that, for (a), 
0, 0, 0; 0, 0; 0g =k(PQSE’ + «E’+rE +p), 
where k is a numerical constant, and «, \ and yw have the 
same signification as in Art. 22. To determine f, let us 
take the particular equation 
a — 5° a’?=0; 
then, making 2,=0, 2,=0, #=5, %=5e, and #,=5e’, 
(e denoting, as in former articles, an unreal cube root of 
unity) we find 
