518 Prof. Young on the Theory of Vanishing Fractions. 
givings about it. The passage I refer to is at page 396, where 
Mr. Woolhouse, in his reasonings on the form °, limits his 
arguments to those comparatively few cases in which the re- 
sults of that form are obtained in such a way, “that no mul- 
tiplication or division by a power of x — a occurs in the pro- 
cess.” If only results obtained under such restrictions as 
these are admitted to come under the second and third prin- 
ciples, then the generality of those principles is of course at 
once given up, and my friend and J are thus far agreed. But 
then so limited a principle of interpretation falls greatly short 
of a general theory; and moreover requires, inits application, an 
acquaintance with the texture of the entire process too minute 
to be generally attainable; it requires, in fact, that we know 
the composition of every multiplier and divisor employed,—an 
impossible problem beyond certain limits. 
At page 399, Mr. Woolhouse enters into a digression upon 
‘the general theory of analytical results,” respecting which 
he considers me to be in error, because in my last letter I had 
said that the fact of the ellipse question, admitting multiple 
solutions, was information which the analytical result was quite 
incompetent to supply; and he observes, “I never before 
heard of the incompetency of an analytical result to afford any 
positive information that an investigation could admit of.” In 
this gratuitous admission of paucity of information upon sub- 
jects in which he so eminently excels, my friend has done him- 
self a wanton injustice. He is too profoundly acquainted with 
all the subtleties of the Integral Calculus, and its applications, 
not to have “heard of” singular solutions, which, though not 
comprised in the resulting integrals which furnish the general 
solutions to certain differential equations, have, nevertheless, 
the property of satisfying the proposed conditions. But a 
more comprehensive view of the results of even common alge- 
bra, would, I think, have induced my friend to withhold the 
remark just quoted. Mr. Woolhouse ascertains the number 
of admissible solutions from “the nature of the problem.” 
By taking a more enlarged view, it would have occurred to 
him that the result might furnish solutions, not only contrary 
to the express stipulations of the problem, but at variance 
with even the original analytical conditions, although these 
may have a much wider range. The results after these “ so- 
lutions étrangéres” are rejected from them, are those from 
among which are to be selected the solutions to the problem. 
In the present discussion it is the connexion between the ana- 
lytical conditions and the analytical results, which is the mat- 
ter before us; and it is, I suspect, from not keeping this in 
mind, that Mr. Woolhouse has been led to say, in mistake, 
