62 
‘¢ Having examined diagrams made some time ago, and 
the calculations connected with same, with a view to ascertain 
the limits of error of this process, it appears that, allowing 
an error equal to 2-10ths of a division of the scale used in 
estimating the sines, the probable limits of error are only 
1-50th part of the numerical sum of the aberrations, a quantity 
which may be considered insensible in practice, and probably 
not one-half of that error, or departure from the true spherical 
figure of a surface, which takes place (and with contrary signs) 
during the polishing, according as the lens, or polisher, is 
upper during the process.” 
Rev. Dr. Graves read a note from Sir W. R. Hamilton, 
in which he stated that he had lately arrived at a variety of 
results respecting the integrations of certain equations, which 
might not be unworthy of the acceptance of the Academy, 
and the investigation of which had been suggested to him by 
Mr. Carmichael’s printed Paper, and by a manuscript which 
he had lent Sir W. Hamilton, who writes,—‘‘ In our conclu- 
sions we do not quite agree, but I am happy to acknowledge 
my obligations to his writings for the suggestions above 
alluded to, as I shall hereafter more fully express. 
<¢ So long ago as 1846, I communicated to the Royal Irish 
Academy a transformation which may be written thus (see 
the Proceedings for the July of that year) : 
D,?+ Dj + D?=-(iDr+jDy + kD:z)?; (1) 
and which was obviously connected with the celebrated equa- 
tion of Laplace. 
“But it had quite escaped my notice that the principles of 
quaternions allow also this other transformation, which Mr. 
Carmichael was the first to point out: 
D?+D,?+ Dy =(Dz-tDz—jDy) (Dz+iDz+jDy). (2) 
And therefore I had, of course, not seen, what Mr. Carmichael 
has since shown, that the integration of Laplace’s equation of 
