232 Royal Society. 



defect in the amplitude of the latter would diminish the value of the 

 ellipticity resulting from the two hy about the -^th part of the 

 whole. If the effect of mountain attraction be as great as the author 

 calculates it to be, (15" - S85 in the northern portion of the Indian 



arc,) the ellipticity would be diminished by - e, and even by as much 



8 



as i £ if the whole Indian arc from Kaliana to Damargida were 



6 

 employed. 



The author then proceeds, first to develope his method of calcula- 

 tion, and then to reduce bis formula to numbers, according to the 

 best data which he was able to collect. 



An expression is first investigated for the horizontal attraction of 

 a prism of the earth's crust standing on a given small base, having a 

 small height, and situated at a given angular distance (measured 

 from the centre of the earth) from the station, A, at which the at- 

 traction is sought. In the cases to which this expression is employed 

 it reduces itself without sensible error to 



M 



— r COS 



G»> 



where M is the attracting mass, « the chord joining its base with A, 

 and 9 the angle subtended by this chord at the earth's centre. 



In applying this expression to the problem in hand, the author 

 divides the earth's surface into lines, by vertical planes passing 

 through at equal angular distances. These lines are further sub- 

 divided by small ciicles having A for their common pole, and in this 

 manner cutting the whole surface into curvilinear quadrilaterals. He 

 then investigates what the law of dissection must be, that is, accord- 

 ing to what law the radii of the small circles must be taken to in- 

 crease, in order that the horizontal attraction of the portion of the 

 crust standing on one of the quadrilaterals may be equal to the pro- 

 duct of its average height and density by a constant quantity, inde- 

 pendent of the distance of the quadrilateral from A. If a and a + <p 

 be the angular radii of two consecutive small circles, there results 



<pcos-(j-a + !<p)_ a constant qU antity = c. 

 sin (*«+#) 



To fix the value of this constant, the author assumes f = — & when 



4 

 <p and a are indefinitely small, which gives c = K7- The above 



equation may then be solved numerically with sufficient approxima- 

 tion. In this manner a table is calculated of the radii of the suc- 

 cessive small circles. 



These distances should be laid down, and the circles drawn, on a 

 map or globe, as well as the lines dividing the surface into lines. 

 Nothing then remains to be done but to ascertain the average heights 

 of the masses standing on the compartments thus drawn. 



The author's paper was accompanied by a plate representing an 



