400 M. BESSEL ON THE ELLIPTIC SPHEROID OF REVOLUTION. 



Its mean uncertainty is = 508'»-7, which is almost equal to its 

 difference from the round number. Hence we see how uncer- 

 tain the length of the metre would have been, even now that the 

 number of measured arcs has been considerably augmented, if 

 its original definition as the 10,000,000th part of the quadrant 

 of the mericUan had been adhered to. Its uncertainty would 

 still amount to at least 0^*0225, a quantity which could only be 

 deemed insignificant in very rough measurements. 



The formulae of which I have to give the numerical develop- 

 ment are the following : 



1. The length of a degree of the meridian of which the mean 

 latitude = <p : 



m=5701l'^-453-284^-851cos2 45 + 0'r-593cos445-0'r-001cos6cf>. 



2. The length of a degree of the parallel : 



j3 = 57l53'r-885 cos 4>-47'^-576 cos 3 4) + 0'r-059 cos 5 <p, 



or if 



sin \^ = e sin 4>, . . . (log e = 8-9110835), 

 then 



logp = 4* 7566845*4 + log cos <p — log cos ^. 



3. Let the radius of curvature in the meridian = r', in the 

 direction perpendicular to it = r", in the azimuth « = r : 



ii= 0"-06314600+ 0"-00031552 cos 2 4* + 0"-00000013 cos 4 ^, 

 ^ = 0"-0G293548 + 0"'00010482 cos 2 4> -0"- 00000004 cos4 cf> 

 or log -^ =8-8025112 .9 + 3 log cos 4/ 



and 



log -fj = 8-7996179 . 6 +;log cos ^, 



— =A + X' cos 2 a, 

 r 



wherein 

 X=O"-063O4O74 + O"-0OO21Ol7 cos 2 <f -)- 0"-00000004 cos 4 <f, 

 X'=0 -00010526 + 0-00010535 cos 2 4. +0 -00000009 cos 4 <p. 



4. Let the distance from the centre of the earth = g, and the 

 corrected latitude = <^' : 



log . g cos cf)' = log . cos $> — log cos \t/, 



log . g sin <p' = log . sin 4= — log cos ^ — 0-0028933 . 3. 



Bessel. 



