24 On Prof. Kelland’s Letter of November 1842. 
no change short of a total alteration of the whole memoir 
can correct it. For instance, the discussion of the particular 
case beginning at the bottom of page 158, which is made the 
foundation of the treatment of the general case, is inapplicable 
except on the supposition of equality of coefficients, for the 
author effects his reductions on that supposition, and that the 
integrals are . 
a =acos(nt—p), B = bcos (nt— p), y =ccos(nt—p); 
and the transference of these, unaltered, into page 163 as the 
proper integrals for the general case, implies beyond the pos- 
sibility of a doubt that the author believed the coefficients to 
be equal in the general case as in the particular case. It is hard 
to understand, after this statement, how we, or any other per- 
son reading the Professor’s Memoir, could have conjectured, 
that in supposing the coefficients equal, we had been led astray 
by ‘‘a misprint or a mistranscription.” 
It is clear, the arguments against what we have advanced in 
our previous communications, are exhausted; we shall there- 
fore consider this letter as concluding our correspondence on 
the subject. 
We are, Gentlemen, 
Your obedient Servants, 
S. EarnsHaw. 
Cambridge, Dec. 2, 1842. M. O’Brien. 
P.S. Dec. 3, 1842.—Perhaps we ought to have noticed the 
Professor’s statement, that he has proved in his Memoir (p. 
180) that the three quantities are unequal, on the equality of 
which ourarguments, in your Magazine for November, entirely 
depend. The Professor certainly deceives himself in thinking 
that he has proved them unequal; for what he has proved is 
simply that v?+v? +v"?= 0; the only legitimate inference 
from which equation is, that v = 0, vo! = 0, uv’ = 0, which is 
precisely what we have proved from the Professor’s equations 
of motion. But the Professor, instead of drawing this infer- 
ence, has imagined from it that v’ is impossible, an inference 
he was not at liberty to make, because it violates the hypothesis 
on which he had effected the reductions and transformations 
in the former part of his paper, upon the truth of which the 
correctness of the equation v? + uv? + vu’? = 0 depends. If 
v' is impossible the integral of his second equation of motion 
ought to have been an exponential; and then the case con- 
sidered at pages 158, 159, is dissevered from the general case ; 
and so the whole memoir would require to be remodelled. 
