Distance traced on the Surface of an oblate Spherotd. 347 
but when the semi-axis of revolution is changed from 1 to P, 
the equatorial semi-diameter will be changed from 4/ 1 + é? 
to P ¥ 1 +e? =a; and the formula will now be, 
e? cos? 
= ! - av 
ds=axds J) ae 
which is identical with the first of the formulze (5) in the Journal 
of Science, because in my notation ds', y, and a stand for 
the same things as do, uw, and e? in M. Bessel’s. The other 
of my formulze (A) is, 
V/ 1+ &sin?y 
Vite ‘ 
LF - / & cosy 
dg=d¢ | i ie aes 
which is identical with the second of the formulse (5). It is there- 
fore certain that the two investigations end in the same results. 
The equations marked (4) in p. 138 of the Journal are no 
more than the equations (5) in p. 139 in a different shape. I 
investigated these equations by giving to the coordinates a cer- 
tain form, which led to them directly without successive sub- 
stitutions, and many intermediate inferences: M. Bessel has 
arrived at the same conclusion by setting out from the usual 
property of the curve of shortest distance on a solid of revo- 
lution, and by a train of reasoning which rests upon proper- 
ties directly flowing from my analysis. How this coinci- 
dence has happened I am not called upon to give any ac- 
count. The date of my solution exculpates me from the charge 
of silently producing formule found by another, as my own 
under a little disguise, in the form of the expression and the 
mode of investigation. 
If we except the general solution of the problem, M. Bes- 
sel’s investigations contain nothing new or of much interest. 
His principal formula (10) in p. 141, is the length of an ellip- 
tic arc expressed in a complicated manner, and requiring in 
practice bulky tables, and the calculation of many subsidiary 
arcs. Let us try with what success the methods we have fol- 
lowed will apply to determine the geographical position of 
places on a given spheroid. 
In the first place we have, 
dg=d¢@'x 
or, 
coslsing Y 1+eé. 
V 1 +? cos*/ ; 
but, from the relation between the arcs z and 7, we likewise 
have, 
cosi = cos Asin wp = 
2K%2 cos 
