208 C. F. GAUSS ON THE GENERAL THEORY OF 



might have a very injurious effect on the results of the elimi- 

 nation*. 



To diminish the unfavourable effect of these circumstances, 

 the number of series of observations from stations well distri- 

 buted over the whole globe ought to be much greater than that 

 of the unknown values, and these should be derived from the 

 observations by the method of least squares. As all the equa- 

 tions are only linear, the process would, it is true, be uniform ; 

 but the extent of the labour, arising from the great number of 

 unknown values and equations, would be such as might well 

 deter the most courageous calculator from undertaking it in this 

 form, especially as the result might be wholly vitiated by the in- 

 troduction either of defective observations or of accidental errors 

 of calculation. 



23. 



There is another mode of proceeding, which, as it is free from 

 a part of these difficulties, appears better adapted for a first trial. 

 We shall develope it in this place without omitting to notice 

 objections to which its application may be liable in the pre- 

 sent state of the inquiry. This method supposes the knowledge 

 of aU three elements at points so grouped on a sufficient number 

 of parallels as to divide them into a sufficient number of equal 

 portions. The numerical values of X, Y, and Z, are to be first 

 deduced from the given elements of the usual form. 



The values of X, Y, Z, are then brought by the known 

 method in each parallel to the form 



X= k + k' cos\ + K' sinX + k" cos2X + K" sin2\ 



+ k'" cos 3 X -f K'" sin 3 X +, &c. 

 r= Z + Z' cos X + Z/' sin X + /" cos 2 X + L" sin 2 X 



+ /'" cos 3 X + L'" sin 3 X +, &c. 

 Z = m + m' cos X + M' sin X + m" cos 2 X + M" sin 2 X 

 + m'" cos 3 X -h M'" sin 3 X +, &c. 



We then obtain as many values for each of the co-efficients k, 

 I, m, k', &c., as there are parallels of latitude under consideration. 



Theory would give in each parallel I = o ; therefore the values 

 of / which result from the calculation fiirnish a kind of measure 



* In such a mode of determination, the effect of these circumstances would 

 be least injurious if the eight points were distributed symmetrically on the sur- 

 face of the earth ; that is to say, if they ijoincided, or nearly so, with the 

 corners of a cube inscribed in the globe. 



