rae ee ae 
113 
and there are five other similar systems of equations, in each 
of which, among the right-hand members, appear two expres- 
sions, differing only in their signs: accordingly, if we diminish 
the roots of the original equation (1) by any one of the six 
quantities, a, a, A; 4, @;, a, the transformed equation will 
have two roots differing only in their signs. 
II.—The second method of solution referred to by Mr. 
Graves, was suggested by observation of the fact that the pro- 
duct of the four quadrinomials, 
w+ ix t+ Py + Pz 
w+ Pa + ty + Pz 
w+ Pat Py + yz 
wtetytz 
in which i stands for V aa A is real, and equal to 
wi—2(y? + 2az)w* + 4y\2° + z*)w + (y°—2az)’—(a* + 2°)? 
Now if we identify this expression with the left hand member 
of the biquadratic equations 
wi + Aow? + Aw + Ay = 0, 
we shall have three equations, from which to determine a, y, 
and z. By the elimination of «and z, we readily deduce from 
these the reduct cubic ordinarily arrived at. 
—_——_—— 
The President made some remarks on the solution of 
equations of the third, fourth, and fifth degrees. 
—_————_ 
[The following Report of the communications made to the 
Academy by Dr. Robinson, on the 25th of April, 1842, and 
the 14th of April, 1845, has been received since the Proceed- 
ings of these dates were printed.] 
VOL, III. k* 
