254 



Archdeacon Pratt on the effect of 



nation of the mean figure. After this I obtain formulae for the mean 

 figures of the Anglo- GaUic, Russian, and Indian Arcs by the same method, 

 each involving the expression for the unknown local deflection of the plumb- 

 line at the reference-station of the arc concerned. I then show that values 

 of these three unknown deflections can be found which will make the three 

 ellipses which represent the three great arcs almost precisely the same. 

 These deflections are not extravagant quantities, but quite the contrary, 

 being small. I infer, then, that the mean of these three ellipses is in fact 

 the Mean Figure of the Earth, and in this way surmount what was the 

 apparently^insuperable difliculty which our ignorance of the amount of local 

 attraction threw in the way of the solution of the problem. The paper 

 concludes with some speculations on the constitution of the earth's crust 

 flowing from the foregoing calculations. 



§ 1 . Effect of Local Attraction on Majpping a Country. 



3. In determining diff'erences of latitude and longitude between places 

 by means of the measured lengths which geodesy furnishes, the method of 

 geodesists is to substitute these lengths and the observed middle latitudes 

 in the known trigonometrical formulae, using the axes of the mean figure 

 of the earth. It might at first sight appear likely that this would lead to 

 incorrect results, as the actual length measured may lie along a curve dif- 

 ferent to that of the mean form. I propose now to show that no sensible 

 error is introduced by following this course, either in latitude or longitude, 

 if the arc does not exceed twelve degrees and a half of latitude, or fifteen 

 degrees of longitude in extent. 



4. First. An arc of Latitude.- — Suppose an ellipse drawn in the plane 

 of the meridian through the two stations, a and b being its semiaxes ; c 

 the chord joining the stations ; « the length of the arc ; r and d, r and d' 

 polar coordinates to the extremities of the arc from the centre of the ellipse; 

 / and I' their observed latitudes ; X the amplitude of the arc ; m its middle 

 latitude : then we have the following formulae, neglecting the square of the 

 ellipticity (e), 



1 2 

 s= — (a-f5)X— - (a—b) sin X cos 2m, 



r=a(l -e sin" /=a(l -e sin^ /'), tan 0=(1 — 2e) tan ?, 

 tan 0' = (1 — 2e) tan I'. 



Now 



c2=rH^"-2rr' cos (0-0') = 2n-'{l- cos {%-^')}^{r-ry 

 =2rr'{l — cos (0—0')}. 

 By expanding the formulae for tan and tan 0', we have 

 = Z— e sin 2^, d'=l'—€ sin 21', 



e-e' = l-l'-e (sin 2/- sin 2/')=:/-/'-2€ sin (/-Ocos 



= X — 2e sin X cos 2m ; 

 1 — cos (0— 0')=l — cosX— 2esin"Xcos2m 

 = 2 sin' ^ { 1 — 2e(l + cos X) cos 2»2} . 



