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, Ui, 02, 03 a^ a^, ag, 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 -^- i^y -\- Pz 



w -|- i^x -j- i'^y -|- i^z 



w -\- Px •\- Py + t'gZ 



w -\- X -{■ y + z 



in which i stands for V ~l, is real, and equal to 

 w''—2(y^ + 2xz)w'' -f 4?/>2 _^ ^2-)^ ^_ (^y-i^2xzf—(x'' + z^. 

 Now if we identify this expression with the left hand member 

 of the biquadratic equations 



w" + A.2W- + K^w 4- A4 = 0, 



we shall have three equations, from which to determine x, y, 

 and z. By the elimination of x 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 * 



