Local Attraction on Geodetic Operations. 



case, the differences of latitudes calculated from the measured arcs of meri- 

 dian with the mean axes, as is done in the Survey operations, will come out 

 free from any effects which local attraction can produce, as that attraction 

 can never be capable of causing so great a distortion in the measured arcs 

 as I have supposed for the sake of calculation. The absolute latitude, 

 however, of the station which fixes the arc on the map will be unknown to 

 the extent of the deviation of the plumb-line caused by local attraction at 

 that place. 



5. Second. An Arc of Longitude. — Let S be the length of the arc, / the 

 latitude, L the longitudinal amplitude or the difference of the longitudes of 

 its extremities, c the chord. Then 



S = L cos /{a + (a — 5) sin^ /} , c— 2 cob I {a -\- (a — b) sin- 1} sin - L. 



z 



When a and b vary, c and I remain constant, but S and L vary. Hence 

 as = aL cos / {a + (a - 5) sin- /} + L cos ? {^a + {U - U) sin" /} 

 = {a + (a-b) sin- /} cos ^ LBL + 2{aa+ (da—U) sin^ /} sin - L. 



By eliminating from these, 



aS= ^L-2tan 1 cos Z + sin^ Z} ; 



2a+(aa— ^6) sin^/=-— — ^ ^ — suppose. 



^ ^ (L-2tan|-L) cosZ 



I will, as before, find the values of la and lb which satisfy this equation, 

 and make la^ -^-Ib' a minimum. 



sin^ I la--\- {(1 + sin- — w}-= a minimum ; 

 .-. {sin^ /+ (1 + sin' iy-}la=n{\ + sin' /) ; 

 • la— (l + sin^/);^ gr_ — ^vc^l.n 



sin^ Z+ (1 + sin^ If ' ~ sin^ /+ (1 + sin^ If ' 



as' 



sm^ (1 + sin' /)' cos'/{sin^/+ (1 + sin'/)'}{L-2taniL}^ 



This is least when cos' Z jsin'^ /+ (1 + sin' /)'} is greatest, or when Z=0 ; 

 then 



la=n, lb = 0, la-lb = n= 



L— 2tan JL 



Now put la - 35 = 13 miles, aS= arc 1" of a great circle =0*0193 mile; 



.-. L-2taniL=0-0193^13=0-0015. 



This shows that L must be small : expanding, we have 



L^ = 0-018, L=0-262 (in arc) =0-262 x57° 3 (in degrees) = 1 5°. 

 We can reason from this, as before, that the differences of longitudes will be 

 accurately found by using the measured arcs of longitude and the mean 

 axes, if the arcs are not longer than 15°. Now arcs of this length, mid of 



