xii SEVENTH REPORT—1837. 
minate, and in employing this very property to decompose certain of 
the conditions into others, for the purpose of avoiding that elevation 
of degree, that would otherwise be the consequence of the elimination. 
The ingenuity of the principle, and the talent displayed in the re- 
searches, are freely admitted by Sir William, who contends that the 
process is valid, as a general and unexpected transformation of equa- 
tions of elevated degrees, though it fails as a method of resolving 
them; and who thus sums up the result of his investigations on the 
subject :—‘ This method of decomposition has, however, conducted, in 
the hands of Mr. Jerrard, to transformations of equations, which must 
be considered as discoveries in algebra ; and to the solution of an ex- 
' tensive class of problems in the analysis of indeterminates, which had 
not before been resolved: the notation, also, of symmetric functions, 
which has been employed by that mathematician in his published re- 
searches on these subjects, is one of great beauty and power.” 
On the very valuable matter contained in the proceedings of the 
Sections time will not permit me to enter, and I must refer you to the 
volume just published. 
In conclusion, allow me, in the name of my respected colleagues and 
of our Liverpool associates, to offer a sincere and hearty weleome to 
the distinguished strangers whose presence confers additional interest 
to this meeting ; and secondly, to congratulate the town of Liverpool 
on the exertions it has made, worthily to receive an Association, which, 
aiming at the diffusion of a general taste for scientific investigations, 
and their application to the improvement of society, seems calculated 
to perform an important part in the future destinies of our country— 
which, as co-operating with all other scientific bodies, and the rival of 
none, but including in its lists representatives from each—-which, distin- 
guished by the freedom of its discussions, the liberality of its assistance, 
and the importance of its recommendations, has been happily charac- 
terized, by an eloquent secretary of a former year, as a Fourth Estate 
in the Realm, and may be aptly designated Her Masrsry’s Partta- 
MENT OF SCIENCE. 
