1862.] Local Attraction. 149 



As noticed in these papers, I conceived also that the difference 

 between the geodetic and astronomical amplitudes might arise, not 

 solely from attraction influencing the plumb-line, but in part from 

 the curvature of the Indian Arc being somewhat different from the 

 curvature of the mean figure of the earth. Geology teaches us, that 

 the earth's surface has undergone changes of level. The surface, 

 therefore, cannot be now an exact spheroid. In this ease the nor- 

 mals at the extremities of the actual arc would include an angle not 

 precisely equal to the amplitude of the mean or undisturbed arc, and 

 part of the errors to be accounted for might, it was thought, arise 

 from this ; the remainder arising from local attraction influencing the 

 plumb-line, and therefore affecting the observed or astronomical ampli- 

 tude. This served to introduce a new element of difficulty. 



4. The ambiguity, however, with which the question was thus beset 

 from all these causes is removed in the Fourth Paper, the last of 

 the sei'ies, published in the Philosophical Transactions of 186J. The 

 following theorem is there demonstrated : — That the length of the 

 actual arc, altered as, its form and position may be by geological 

 changes, is nevertheless sensibly equal to the length of the mean or 

 undisturbed arc. Hence, if we calculate the amplitude by using the 

 measured length of the arc, and the mean axes, as is done in the 

 Survey, it will come out the mean or undisturbed amplitude. The 

 consequence of this is, that the relative position of places laid down 

 on a map from geodetic operations is correct, and free from all sensi- 

 ble error arising from local attraction, from whatever causes local 

 attraction may arise. 



This is a most important practical result, and frees the Survey 

 operations from a doubt which has attached to their high scien- 

 tific accuracy, ever since it has been discovered that the influence 

 of the Himalayas and of the ocean is so considerable, and that 

 variations in the earth's crust below may have an important disturb- 

 ing effect. This theorem, moreover, gives us a direct means of 

 estimating at once the difference of local attractions, and of local 

 deflections caused by them, at the extremities of an arc. For the 

 difference is precisely equal to the quantity by which the astronomi- 

 cal amplitude differs from the mean or undisturbed amplitude found 

 as above described. 



5. There is only one desideratum remaining ; but one which I 



