618 REPORT— 1888. 



6. On the Errors of the Argument of Statistical Tables. 

 By Joseph Kleiber. 



The object of the paper is to show that statistical tables and generally tables 

 giving the result of observations on two covariable quantities are not invertihle, the 

 values of the function not corresponding exactly to the values of the argument as 

 indicated in the table. The correction to be applied to the argument for the inver- 

 sion of the table may be calculated by very simple formulte. If we have a table 

 giving a series of values of y for equidistant values of x, and we take for unity the 

 distance between two consecutive values of x, thea the true value of an argument 

 Xi is represented by the integral 



I xf(x)dx .... (1) 



f(x) being the probability curve of x, which may be approximately found by con- 

 sidering the distribution of the number of cases, when x had given values x^, x ^ . . . 

 If we have, for instance, ??i cases of x being equal to .r^, and the total number of 

 observations = n, then 



ni = n[' '' f{x)dx .... (2) 



From a series of values of w a parabolical expression for /(.i) may be calcu- 

 lated by means of the formula (;!), and this expression being substituted in (1) 

 gives with sufficient accuracy the required correction. 



Ex. — In the simplest case of three observations being given, corresponding to 

 the arguments y = — 1, 0, 1, and the number of cases of .r = — 1, 0, 1, being respec- 

 tively »i_i, Wq, «j, we find for the correction of the value .r = the formulae 



Imj— n_j 



SMj + ?i(j + n_i 



if we have for y(.r) a linear expression, and 



1 «!— n_i 



if /(.r) is represented by a parabola of the second order ; and similar formulse may 

 be found for higher orders of parabolse and larger numbers of arguments. The 

 paper will be prepared for publication in the ' Philosophical Magazine.' Examples 

 of application to meteorological tables will be given in the ' Meteorologische 

 Zeitschrift.' 



7. On Geometry of Four Dimensions. By Edward T. Dixon. 



In this paper the author begins by pointing out that the principles of geometry 

 of four or any number of dimensions might be worked out analytically, even if 

 they could not be interpreted. He then proceeded to explain how it was possible 

 either to conceive a fourth independent direction or to graphically interpret 

 geometry of four dimensions without this conception by regarding the density of a 

 solid as a geometrical dimension analogous to length. Thus mass became a 

 geometrical dimension as well as length, area, and volume, and any equation in 

 four variables might be graphically represented by a solid body of varying 

 density. 



8. A Suggestion from the Bologna Acaaemy of Science toioards an agree- 

 ment on the Initial Meridian for the Universal Hour. By Dr. C^s. Tondini 



DE QaARENGHI. 



In a letter dated June 18, 1879, to the Secretary of State for the Colonies, 

 and relating to the choice of the initial meridian for the universal hour. Sir G. B. 



