CALIBRATION OF A GLASS TUBE. 311 



A second column of double length 2/, which is measured in 

 the same way, by placing its ends successively near the principal 

 points, will give n - i equations of the form 



A column of triple length 3/ will give n - 2 equations, 



" 



Continuing in this way with increasing columns, the column of 

 length equal to (n- 2) I will give 3 equations, 



and the last column two equations, 

 / 2 y-u o-%-i + ^ 



We are thus led to a system of n - --- i equations with 



2n unknowns, or rather 2(ni) unknowns, since a and a n are 

 zero, which determine the corrections of all the principal points 

 with the same accuracy. The particular symmetry of this system 

 enables us to resolve equations under different symmetrical forms. 

 It will be more correct to consider the different values of Aj as 

 the unknown, but the probable error is not sensibly increased if 

 we eliminate these quantities by the subtraction of two consecutive 

 equations of each group.* 



* M. THIESEN. Carl. Reptrtorium, Vol. XV., pp. 285 and 678. Munich, 1879. 



