31 6 E. D. Preston — Deflection of the Plumb-line, etc. 



of the ocean very much more dense than the land. The depth 

 of the ocean is about 10 1 00 of the earth's radius, and the vol- 

 ume of the prism would therefore be decreased by its -3-^3-rd 

 part. Hence, if we have the same amount of matter, the den- 

 sity should change by such amount This is entirely inade- 

 quate to counteract the effect of the light matter above, or ma- 

 terially change the deflections brought to light by the zenith 

 telescope or by the pendulum. 



It has been found that variations that arise from hidden 

 causes under the Himalayas are two or three times as great as 

 those that arise from the mountains themselves. The compu- 

 tations on which this result rests assume Young's rule, and 

 take the ratio of the surface density to the mean density to be 

 one-half. Besides, it is well known that the rule supposes the 

 matter lying under the station to be a plain of infinite extent. 

 The same rule has been applied to mountains, supposing them 

 to be either cones, cylinders or the segments of a sphere. Ev- 

 idently, in these latter cases, the error is in the direction of 

 making gravity on top of a mountain too great, because the 

 rule corrects additively for too great an amount of matter. 

 In fact, to suit Haleakala, the constant factor in Young's for- 

 mula should be changed from 1*25 to l - 36, and we should have 



?L = (l - * 1) 



g« V 3 rj 



91 



9, 

 where the factor j- of the ordinary formula is here replaced 



by f.. 



This increase of the factor seems at first a paradox — but we 

 must remember that the essential tendency of gravity is to di- 

 minish as we rise, and that any correction for matter is a posi- 

 tive quantity which must be added to this negative correction. 

 Hence, for a cone, which contains less matter than the plain, 

 the decrease of gravity should be greater, or the factor which 

 corrects for the matter should be greater. 



In fact, the factor that corrects for an infinite plain has only 

 to be multiplied by a quantity depending on the cosine of the 

 vertical angle of the cone to make it applicable to this figure. 

 Supposing the cone to have a vertical angle of 180°, the two 

 formulae coincide, as they should do, since in this case our 

 cone becomes an infinite plain. 



Using the new formula 



* = 2*[l -l d - (l-cos/3)! 

 g r L 4 A J 



to get an improved mean density we arrive at 2*8 — the de- 

 crease of gravity shown by the pendulums indicating a ratio of 

 exactly \ for the two densities : gravity is first corrected for 



