346 Mr. Ivory on the Properties of a Line of shortest 
cording to the formula (F) the arcs x and y ought to be more 
nearly equal than they are found to be. I have likewise com- 
puted the difference of longitude ¢ directly from the azimuth 
by means of the formula (G), and have found it equal to 
5° 4! 3-32, or 0-46 less than it should be, which agrees with 
the remark just made. 
I shall not prosecute this subject further at present. It 
would be interesting to investigate the general case of a geo- 
detical line directed in any angle to the meridian ;_ but it would 
occupy too much room. ‘The relations of all the quantities 
concerned in the problem have, in the foregoing analysis, been 
expressed by formule so simple and manageable, that there 
can be little difficulty in the investigation of any point that can 
occur in practice; and it is in this that I conceive the advan- 
tage of the solution I have given to consist. 
Since my last communication on this subject, the 41st number 
of the Quarterly Journal of the Royal Institution has appeared, 
which contains some investigations of M. Bessel relating to 
the curve of shortest distance on a spheroid of revolution. It 
is extremely remarkable that M. Bessel’s general solution of 
the problem is exactly the same with that which I published 
in this Journal for July, 1824*. By saying this I mean, not 
that every step of his investigation is the same with mine, but 
that the same view is taken of the problem, and the ultimate 
formule obtained, are not a jot different from those which I 
have given. The two formule marked (5) in p. 139 of the 
Journal of Science, are identical with the two marked (A) in 
my solution, the apparent difference existing only in the nota- 
tions. Although this is so plain as to require only to be no- 
ticed, yet ina case of this kind it may not be improper to 
prove incontestibly the exact coincidence of the expressions. 
Now one of my formulee (A) is this, 
ds=ds' 71+€sin*y, 
which belongs to a spheroid of which the semi-axis of revolu- 
tion is unit; and if the same semi-axis be of any other mag- 
nitude P, it is evident that we must write = for ds, and then 
we shall have, 
ds=Pxds 7W1+é€sin’y: 
put 1—cos* for sin*; then 
Pia Poe et RUE of de SOE 
l+e 
* It is proper to observe that I have no knowledge of M. Bessel’s writ- 
ings on this subject, except from the Journal of Science. 
but 
