AT THE SURFACE OF THE EARTH. 691 



28. Let us return to the quantity F^ of Art. 19, and consider how tlie attraction of the earth's 

 irregular coating affects the direction of the vertical. Let / be the latitude of the station, which 

 for the sake of clear ideas may be supposed to be situated in the northern hemisphere, sr its 

 longitude west of a given place, f the displacement of the zenith towards the south produced by 

 the attraction of the coating, tj its displacement towards the east. Then 



p_ _}_^ _secl dV^ 



Ga dl Ga d'gr 



1 dV seel dV 

 because - — y and ~ are the horizontal components of the attraction towards the north 



Qui (I (I'^j- 



and towards the west respectively, and G may be put for g on account of the smallness of the 

 displacements. 



Suppose the angle ;;^ of Art. 22 measured from the meridian, so as to represent the north 

 azimuth of the elementary mass ^a- sin \|/d\^dj('. On passing to a place on the same meridian 

 whose latitude is I + dl, the angular distance of the elementary mass is shortened by cos v . dl, and 



therefore its linear distance, which was a chord \|/, or 2o sin — , becomes 2 a sin i - o cos — cos v dl 



o 2 o A. ' ' 



Hence the reciprocal of the linear distance is increased by — cos — cosec'— cos v . d/, and therefore 



4a 2 2 '^ 



the part of V^ due to this element is increased by l^a cos^ — cosec — cosy .d^dydl. Hence we have 



dV. 



cos — cosi^ 



ar^ arc 2 "■ „ 



sin — 



Although the quantity under the integral sign in this expression becomes infinite when -J/ 

 vanishes, the integral itself has a finite value, at least if we suppose }> to vary continuously in the 

 immediate neighbourhood of the station. For if I becomes }>' when j^ becomes y + ir, we may 



replace ^ under the integral sign by ^ — ^', and integrate from y^ = Q Xo y_ = -k, instead of integrating 



^ _ ?' .^ 



from y = to Y = 27r, and the limiting value of — ^ when \J/ vanishes is 4 , which is finite. 



. v a \1/ 



sin— ' 



2 



To get the easterly displacement of the zenith, we have only to measure y from the west 

 instead of from the north, or, which comes to the same, to write \-V - for y^, and continue to 

 measure y from the north. We get 



seci . — = — /"/"cos'— cosec — sin y.^dv/^dy (41) 



dsr 2 ■'•' 2 2 -^ ^ '^ ^ ' 



29. The expressions (40) and (41) are not to be applied to points very near the station if ^ 

 vary abruptly, or even very rapidly, about such points. Recourse must in such a case be had to 

 direct triple integration, because it is not allowable to consider the attracting matter as condensed 

 into a surface. If however vary gradually in the neigh Imurhooi! of the station, the expression 

 (40) or (41) may be used without further change. For if we modify (lo) in the way explained in 



the preceding article, or else by putting the integral under the form /„"/„-' cos' — cosec ^ cos j( 



(S - l,)d\\,dy, where ^, denotes the value of I at the station, we sec that the part of the integral 



