144: W. D. Lambert — Mechanical Curiosities 



Values of the Second Derivatives of the Gravity Potential 



Observed near Padua, Italy. 



[Unit = one 10-^ C. G. S. Unit.] 



No. of 



9"Y 9'Y 



9'y 



9'Y 



9'Y 



Station 



9y' 9x' 



9x9y 



9x 9z 



9y 9z 



I 



+ 14.62 



— 11.69 



— 4.91 



— 41.19 



II 



+ 8.39 



+ 6.78 



— 5.98 



— 35.26 



III 



+ 17,31 



— 1.49 



— 14.53 



— 21.90 



X 



+ 41.58 



— 0.26 



— 14.09 



+ 4.59 



XI 



— 15.28 



+ 4.74 



+ 0.18 



— 3.07 



XII 



+ 37.93 



— 1.32 



+ 1.16 



— 13.47 



XIII 



+ 10.92 



— 2.14 



— 0.36 



— 3.12 



XIV 



+ 5.74 



— 6.15 



+ 16.65 



— 2.94 



These values are not rare exceptions due to very abnor- 

 mal local conditions. It would be easy to find still more 

 erratic-looking observations.^^ The specimen given may 

 be taken as fairly typical. Nor are these values the 

 results of observational blunders. Repeated observation 

 in the same spot reproduces consistently the same value 

 and the results check with pendulum observations and 

 with the observed deflections of the plumb line deduced 

 from a combination of astronomic and geodetic opera- 

 tions. The details of the connection between these three 

 kinds of results would take too long to explain here.^"^ 

 These peculiar-looking results are simply a numerical 

 illustration of what the text-books tell us — that while the 

 potential function and its first derivatives change contin- 

 uously in the passage through discontinuities in density, 

 the second derivatives are discontinuous whenever the 

 density changes discontinuously. A surface of discontin- 

 uity in density, namely, the uneven surface of the ground, 

 is necessarily near the instrument at all times. The 

 observations are not taken at this surface itself, hence the 

 derivatives obtained are not strictly discontinuous in 

 value, but merely vary with great rapidity in a way inti- 

 mately related to the irregularities in the density and 

 shape of the matter near the surface. 



^^ See for instance footnote on p. 139, Art. D., p. 435, in which an obser- 

 vation is given in the canyon-like valley of Cimabanche in the mountains 

 near the former Austro-Italian frontier, approximate latitude 46° 35', 

 approximate longitude 12° 10' E. of Greenwich. 



^* Footnote as above under C. pp. 325-332, or under G. p. 59, or under H. 

 p. 53. 



