' in reply to the Observations of Professor Young. 399 
Young’s views of the ellipse question. It may, however, be 
worth while to take the opportunity of adding a single remark on 
an erroneous principle which he appears to entertain regarding 
the general theory of analytical results. I never before heard of 
theincompetency of an analytical result to afford any positive in- 
formation that an investigation could admit of. It is plain that 
the original equations, which express the analytical conditions 
of a problem, cannot include any extraneous conditions with 
those expressed in the enunciation, and that they must there- 
fore comprehend, in their analytical results, every solution 
that the problem is capable of receiving. The equations, how- 
ever, may not include certain other implied conditions, de- 
pendent on the peculiar nature of the inquiry, and therefore 
may yield some additional solutions incompatible with the 
conditions so implied. For instance the nature of a problem 
may be such as to exclude from the results not only imaginary 
values but negative values and values which fall beyond cer- 
tain limits, though they will be unavoidably comprehended in 
the analytical solution. ‘The exclusion of inadmissible solu- 
tions, therefore, rests with the nature of the problem and not 
with the forms of its analytical conditions. It is hence evident 
that Professor Young involves himself ina palpable incon- 
sistency, when he arrives at the fact of the ellipse question 
admitting multiple solutions, by an examination of the origi- 
nal analytical conditions, and at the same time alleges that the 
analytical result is quite incompetent to supply that informa- 
tion; for the true analytical result must necessarily present 
every solution capable of satisfying the analytical conditions 
from which it has been deduced. If we refer back to the na- 
ture of the problem, as originally presented, which is the pro- 
per source of rejective information, we perceive that the only 
condition it imposes on the results is the limitation which re- 
quires the coordinates xy to fall within the bounds of the 
ellipse, or of the circle that represents it in the indeterminate 
case. 
I have thus unreservedly enumerated the principal reasons 
on which I found my sincere and firm conviction of the incor- 
rectness of the various statements contained in Professor 
Young’s letter. To avoid the possibility of being misunder- 
stood, I have also given a concise analysis of the most impor- 
tant of the principles maintained in my essay ; and, in conclu- 
sion, I may be permitted to add, that instead of their being 
*condemnatory of conclusions which, in the works of our 
ablest modern analysts, wear all the aspect of mathematical 
certainty,” they establish the truth of those very conclusions 
on a firmer and more intelligible basis,—that instead of. the 
