TRANSACTIONS OF THE SECTIONS. 



11 



Example 1. Let the degrees of the coefficients be 



ly 3, 4, 4, 4, 3, 2, 1, 0, 



the equation being of course of the 8th degree. Then constract the following figure, 

 in which the ordinates are proportional to the degi-ees of the coefficients : 



The diiferences between the degrees of the coefiicients are 



2,1,0,0,-1,-1,-1,-1; 



and consequently the degrees of ^{x) in the particular integrals of the equation 

 will be 



3,2,1,1,0,0,0,0; 



so that in the general solution there will be 



One integral of the form eai3+^x2+ya;Q^ 



One 

 Two 

 Four 







-+^*Q, 



of a pui-ely algebraic form. 

 Example 2. Let the degrees of the coefficients be 



1, 3, 3, 4, 4, 0, 2, 3, 2, 0, 

 the equation being of the 9th degree. Then form the figure 



where, after bridging over the re-entering angles, the differences of the degrees are 



-2; 



2 1 1 -1 -^ -1 -1 

 '''2'2' ' 3' 3' 3' ' 



and consequently the degrees of the exponentials ^(x) will be 



q33,222^, 



'^> 2' 2' ' 3' 3' 3' ' • 

 About the degree — 1 there is a difficulty ; but the author suggests that the nega- 

 tive index arises from an accidental cancelling of the highest power of x in Zo, and 

 that it may probably be replaced by zero. 



On the Reduction of the clecadic Binary Quantic to its Canonical Form. 

 By WrLLiAM Spoxtiswoode, M.A., F.B.S. 

 Professor Sylvester has shown that the quantic 



may be reduced to the form 



Ui2"+Uj«+ —tin^»+AVu^tt.-^ . . Un, 



in which A is a constant, and V a covariant of the product Mj, u^ . . tin, satisfying a 

 certain differential equation. In applying his method to the 10th degree, the greatest 



