16 OLDHAM: THE STRUCTURE OF THE HIMALAYAS, ETC. 



neglected, the effect of the compensation, at the station S, is re- 

 presented by the formula 



m 

 D' = ,-o sin 2 a cos a 

 h 



where D' represents the deflection produced at S, 



m, the mass whose effect is supposed to be concentrated at C, 

 h, the depth of C below the level of 8, and 

 a, the angle of depression of C at A. 



This expression has a limiting value of zero when a is 0° or 

 90° and attains a maximum when it has a value of about 50° 45', 

 or when the depth h is about 1*4 times the distance r. If this 

 maximum effect, and the distance at which it is produced, are 

 both expressed as 1*0, the proportion of this maximum effect which 

 will be observed at other proportionate distances is given in Table i . ' 



Table 1. — Relation between distance and effect of the attraction of an 



under-ground mass. 



Distance. 



Depression. 



DeHection. 



•1 



86° 



4' 



•15 



•2 



83° 



22' 



•30 



•3 



80° 



2' 



•44 



•4 



76° 



39' 



•57 



•5 



73° 



14' 



•69 



•8 



69° 



4-4' 



•79 



•7 



66° 



10' 



•88 



•8 



62° 



30' 



•95 



•9 



58° 



42' 



•98 



1-0 



54° 



45' 



1-00 



11 



50° 



36' 



•98 



1-2 



46° 



10' 



•94 



1-3 



41° 



23' 



•85 



1-4 



35° 



7' 



•70 



If) 



28° 



52' 



•53 



1-6 



21° 



r 



•31 



1-7 



7° 



38' 



•04 



1 This table has a further utility in that it may be applied to the effect of any defect 

 or excess of mass at any depth below the level of the station, where the distances involv- 

 ed do not introduce the necessity of considering the curvature of the earth's surface, 

 and where the dimensions of the mass are such that it may be regarded as centrobaric 

 at all the distances involved. In the case of more extended masses the effect is the 

 sum of the effects of all the separate small masses of which it is composed, and this effect 

 would usually diminish the ratio between the distance of maximum effect and the mean 

 depth of the mass, but not reduce this below equality. It is unnecessary for the present 

 purpose to treat this matter in further detail ; it is sufficient that the mean depth of 

 the centre of such a mass will lie somewhere between 1 and 1 "4 of the distances 

 between the positions of maximum and zero effect. 



[ 164 J 



