182 



EEPORT 1882. 



let Uj.,. be the number entered in the r*'^ row of the h^^ column (the 100, 

 which for shortness has been omitted, being of course included). 



The symbolical representation of Table XXI., with the Vs, &c., intro- 

 duced, is therefore 



where 



M = Yl 



+ V, + +V,o_H, +H, + +H 



in 



10 



10 



Now, von Oettingen proves (loc. cit. p. 9), that 



^U,,.= H,. + V,-M-U,,. 



An alternative plan is, however, better — viz., to make use of Table 

 XIX. instead of Table XXI., inasmuch as the numbers employed are 

 smaller. Let /.i be the mean of the initial points, and let Ji,., v^., in and %^ 

 have the same significations with respect to Table XIX. as the corre- 

 sponding large letters have to Table XXI. Then 



H,. = i, + h,., V/, — ^L + i\., M = n + m, U,.,. = i, + u,,,. 



.'. by substitution in the above formula, 



fl> (U;,,) = h,. + Vk - »«' - «<■-■• 



The values of all the four quantities on the right-hand side of this 

 equation are given in Table XIX. 



Table XXIII. contains the values of v^ — %.,., and in the last column 

 of h,. — m. 



Table XXIV. is obtained by adding the values of h^ — rti to those of 

 V)i — V,.,. — e.g. in Column I. 176 — 191 = — 15, and so on. 



Table XXIV., therefore, contains the corrections according to the 

 formula for the scale readings given in corresponding situations in 

 Table XXI. A curve is then drawn, with the scale divisions in Column I., 

 Table XXI. as abscissae, and the corresponding corrections in Column I, 

 Table XXIV. as ordinates, and similar curves are drawn for each of the 

 other threads. 



They are shown in fig. 1, Plate II., drawn in continuous lines, and 

 numbered to correspond with the column and thread from which Ihey 

 are deduced. The scale of the curves in the plates is one-half of that 

 actually employed. The mean of the ordinates of these curves for every 

 degree above the highest initial point is then taken, and a curve with 

 these quantities as ordinates is drawu, tluis forming part of the^^rs^ mean 

 cm-ve. It is shown in fig. 2, Plate II., drawn in a continuous line. It will 

 be observed that the fii-st correction curves do not extend to the lower 

 and of the scale. The corrections for this are obtained from the initial 



