491 
GENERAL SOLUTION OF ALGEBRAIC EQUATIONS. 
that from the very nature of their con- 
nexion with the original equation, they 
must fail in giving its solution, where it 
only rose to the fourth dimension, be- 
cause he showed that this would involve 
the solution of an equation of the sixth 
degree, as a preliminary step. Equa- 
tions, however, of this degree had been 
long solved, and it was only, therefore, 
in connexion with the generality of Mr. 
Jerratd’s method, that its failure, as re- 
garded them, was of any consequence. 
He then proceeded to give a similar de- 
monstration of its failure, as regarded 
equations of the fifth and of tiie sixth 
degree; during his discussion of this 
step of his demonstration, he took occa- 
sion to show, that Mr. Jerrard’s method 
had succeeded in reducing equations cf 
the fifth degree to tables of double en- 
try, — a discovery, ui)on the value of 
which he enlarged considerably, and 
highly eulogized and complimented the 
author; insomuch, that he stated that 
if the method had accomplished nothing 
but this alone, Mr. Jerrard would have 
received the congratulations of the scien- 
tific worlJ. He then proceeded to 
show, that unless the index of the equa- 
tion reached as a minor limit the num- 
ber seven at least, a certain intermediate 
equation, concerned in the elimination, 
would be met with, along with a multi- 
ple of it, which, therefore, would not 
give a number of distinct results suffi- 
cient to complete the uliminations ; but, 
beyond that degree, he stated that he 
had satisfied himself, that Mr. Jerrard’s 
method would afford solutions of equa- 
tions, which, even if they should, from 
their complexity or other causes, be use- 
less to the j)ractical or merely arithme- 
tical algebraist, yet to those engaged in 
prosecuting inquiries involving purely 
symbolic alg-bra, he felt confident they 
would afford facilities and general me- 
thods of investigation, hitherto almost 
un'ooked for and unexpected. 
Mr. Babbage comjfiemented Sir W. 
Hamilton upon the very lucid exposition 
which he had given of a subject which 
he characterized as bordering upon the 
very extremest limits of human know- 
ledge, and congratulated Mr. Jerrard upon 
the success with which he had contrived 
so effectually to distinguish between the 
symbols of operation and those of quan- 
tity, in expressing the results of elimi- 
nation. Engaged, as it was well known 
he was, in a branch of practical numeri- 
cal science, he could not suffer himself 
to be supposed to look with indifference 
upon a discovery which, if it should 
even fail in affording any practically 
important assistance to his particular 
branch, must yet be admitted to afford 
the strongest promise of advantage to 
the more purely al)stract branch of alge- 
braic investigalion — Professor Peacock 
observed that during the progress of the 
discussion of this question he had not 
failed to remark the many advantages 
which must result to algebra from Mr. 
Jerrard’s method, from the collateral 
improvements to which the i^rosecution 
of his princij)al'o!)ject had led, partly in 
snggusting new and hitherto unex- 
])lored methods of elimination, and part- 
ly by leading to a notation, which so 
clearly distinguished between the marks 
of quantity and the observations and 
changes which were to be resorted to in 
reference to them ; but as to the result 
itself, he need characterize it no higher, 
when he added, that it was an advance 
in ihe science, vvhich it did not appear 
that the celebrated La Grange had ever 
contempleted, and which was not ap- 
jwoached by the result of Stcliern- 
hausen. — Mr. Frend regretted to differ 
from Mr. Babbage ; but to him it ap- 
peared that this subject, instead of rang- 
ing on the very outside verge of human 
knowledge, on the contrary, lay at the 
very entrance of its portals. 
Professor Phillips then read his Re- 
port of the Experiments instituted with 
a view to determine the Inferior Tem- 
perature of the Earth. 
Mr. Phillips stated, that this subject 
had for a long period engaged the 
anxious attention forscientific men, both 
at home and upon the continent ; that the 
most accurate as well as numerous ex- 
periments indicated a decided elevation 
of tera})erature as a more depressed sta- 
tion below the earth's surface was at- 
tained ; even when the depths descend- 
ed to were small, this elevation of tem- 
perature became large enough to arrest 
attention ; in fact, the temperature of 
the air, of the water, of the rocks, and of 
the soil, was found to augment as we 
descend. But, in order to ascertain 
if possible, what portion of this heat 
arose fronri, or was connected with, an 
elevated temperature of the internal 
parts of the globe, as well as to ascertain 
whether the causes of these were local 
or universal, and, if possible, to arrive 
