Report on the Solar Eclipse Expedition to Sahdol. 17 



we see that w is constant when the arc PP is constant, i.e., along the 

 arc of a small circle described with centre P and radius PP . 



The bounding lines of the totality belt given us by the approxi- 

 mate formulae of the 'Nautical Almanac Circular,' are thus portions of 

 small circles of the sphere. 



Now, suppose we have formulae given for the neighbourhoods of 

 two points B and b on the central line ; for which P and p are the 

 centres of the approximate loci of equal totality. Then the approxi- 

 mate formulae for B will give us as the northern boundary of the 



totality zone the circular arc AD, touching the true line at A, but 

 falling south of it elsewhere ; and the approximate formulas for b 

 will give us the arc ad. Thus at the intermediate point D, both 

 formulae give errors in the same direction; and unless the points 

 B and b are tolerably close together we cannot get a good prediction 

 for intermediate points by simple interpolation. 



The true lines Aa, B6, Cc, are the envelopes of such circles as AD, 

 BE, OF, as we travel along the central line, P travelling along the 

 path P j?o in correspondence. 



It is to be noted further, that the approximate formulae give a 

 constant duration of totality along the line BE, which is the approxi- 

 mate central line (given by sin ta = 1) : whereas the duration gener- 

 ally changes as we go east or west. But interpolation between 

 results for B and 6 would probably give the means of allowing for 

 this change. 



If instead of the duration of totality, we take the time of one of 

 the contacts, the approximate loci are no longer circles, but curves 

 of the form. 



and the geometry is less simple ; but these curves will have the true 

 line for their envelope just as the circles did. 



Now, if the contact of the circles or the curves with their envelope 

 is of the second order interpolation becomes possible, for the circles 

 cross the envelope and thus the error introduced is + ^to the east 



VOL. LXIV. c 



