ON THE GAUSSIAN CONSTANTS FOR THE YEAR 1829. 17 



having added each of these contributions to the similar one among those 

 equations which H. Petersen has stated to represent all the magnetic elements 

 measured by Erman in 1829*, we had to solve the so completed expressions 

 according to the sought-for corrections, A^'", A(/''', AA'"'. . . . AA*'*. 



On the other hand, instead of such indifferent aggregation of all new 

 material to all the old, we had, according to the second method, to make 

 a proper abstract of each of the two classes of data, and then to derive the 

 sought-for values of constants from equations founded only on these abridged 

 materials. 



But as the most probable determination of the Gaussian constants is 

 evidently only obtained by observations at points symmetrically situated all 

 over the earth's surface and being all of equal weight (that is to say, reliable to 

 an equal extent), the beforementioned method jjroved to be doubly imperfect. 

 Indeed the material for the said former calculation of H. Petersen consisted in 

 GlOmagnetic elements, which corresponded to 650 direct observations executed 

 along a line round the earth of 8100 German miles. The three data for the 

 magnetic determination of a point, therefore, were to be found all over this 

 line at an average distance of 37'4 German miles, or of very nearly 2°-5 of 

 the equator, whereas when those points for which magnetic elements had 

 now to be added were counted in their succession on parallels of latitude or 

 on any other lines round the earth, there appeared everywhere a much 

 scantier distribution, which on an average did not exceed a sixth or a seventh 

 of what it was for the former calculation. On immediate addition of the 

 former suras of final equations to the corresponding new sums, the resulting 

 new values of Gaussian constants would therefore have been influenced to an 

 exceedingly larger extent by the magnetic character of one almost linear 

 tract of the earth's surface, than by all its remaining parts. To compensate 

 such vicious preponderance, we might, before adding the two sums, have 

 multiplied each of them by a number inversely proportional to the frequency 

 of its elements. But this proceeding supposed, in order to be right, that all 

 constituent observations were of equal weight, while in our case we must 

 own, on the contrary, that the probable errors of the newly added elements 

 surpassed those of the formerly observed ones in a considerable though rather 

 indefinite proportion. Indeed by separate comparisons of some of the new 

 and of the old observations with others of their respective classes, the new 

 seemed upon the whole in less accordance, partly of course in direct con- 

 sequence of the manner in which they were made, partly because of their 

 having been reduced to 1829 by a method which, for all our care, was but 

 an approximation to reality. 



We have avoided these difficulties by choosing the second of the above- 

 mentioned modes of operation and by prosecuting it as follows : — 



Out of all stations for which the three rectangular magnetic components, 

 as in 1829, had become known, either by direct observation or by our 

 reductions for secular changes, we selected those which are nearest to ten 

 parallels of latitude between i6=23° and zf=165°, and at the same time to 

 the one or the other of nine equidistant points of every one of these circles. 

 Having then concluded out of the results for these stations the 270 elements 

 that belong to the 90 predetermined points, these latter values were ex- 

 clusively introduced into the like number of our primary equations, which in 



* As published in the Eeport of the Eighteenth Meeting of tlie British Association, held 

 in IS4S, tables facing p. 08, under "Final Equations i'or the corrections of the Gaussian 

 constants from 610 magnetic elements.'' 



1.S72. c 



