200 EEroET— 1882. 



In Table LII., the last three columns are in virtnal agreement, while 

 the second difiPers from all of them in that the correction in the neighbour- 

 hood of 108° is too great. This is exactly what would occur on the 

 assumption that von Oettingen had not given sufficient weight to the 

 initial jioint curve. 



The maximum error would occur in the neighbourhood of the point 

 where the initial and upper point-curves first overlap, as there, owing to 

 the small number of the upper point-curves, the errors introduced by 

 them would be most important. The nature of the error would be such 

 that the correction carves would be deflected too far in the direction of 

 the upper point-curves, and finally it Avould be eliminated by a sufficient 

 number of approximations. 



All these conditions are fulfilled. The overlap begins at 106. At 

 that point there is a sudden increase in the difference between the first and 

 second approximations, which is a maximum at 108°. The upper point- 

 curves lie above the initial point-curve ; and the second apj^roximation, 

 given by von Oettingen's method, lies above the third. The third 

 approximation agrees very closely with the second and third approxima- 

 tions obtained by Professors Thorpe and Riicker's method, which are them- 

 selves in virtual agreement. 



The upper part of von Oettingen's curve is, however, for the most 

 part above that obtained by the other method, and this error is no doubt 

 due to the less rapid approximation obtained by ranking the initial point- 

 curve as equal to one of the upper point curves only. 



On these grounds, therefore, the curve calculated, on the supposition 

 that the initial point-curve is equivalent in value to ten of the upper 

 point-curves, will be taken as the standard. 



(35) Only one other point with regard to Bessel's method requires 

 investigation, viz. the error introduced by the measurement of the thread, 

 not at, but near, the principal points. Let the true lengths of the divi- 

 sions on which the upper and lower ends of a thread lay be 1 -H c?„ and 

 1 + di. Let y be the fraction of the division in which the lower end lay 

 by which that end was distant from the principal point. Had the thi-ead 

 been pushed back through this distance its length would have been 

 increased by ?/((?„ — d^. 



All the numbers in Tables XX. and XXI. must therefore be increased 

 by this amount. Assuming y to have a mean value y throughout, and 

 inserting this correction in the formula on p. 182, it is easily proved that 

 the added correction of U^^ is 



y[^^rd. - d,^ - 1 S,c7„ _ _!_ S„ S.cZ„}, 



where S,. d„ and S^, d,,, are the sums of all the values of d„ in the r"^ row, 

 and the ¥^ column respectively, and S^. S,. c?„, is the sum of all the values 

 of c7„. 



The correction for the r^^ initial point is increased by y {S,.(7„ — d,] . 



On taking out the values of d^ and d^ from the curves in fig. 1, Plate 

 III., it is found that the largest correction for any upper point is 

 0-002y, and for any initial point Q-OOoy, and since the value of y is about 

 0-3, the error introduced cannot exceed 0°001. The mean curves may, 

 therefore, as far as this error is concerned, be taken as correct to 0°001, 

 and it has therefore been neglected. 



