OF ECCENTRICITY AND GRADUATION. ' 31 



greater than t, there is reason to suspect that the graduation is sensibly- 

 imperfect. The probable error of any angular measurement, due to the 

 errors of the vernier corrections, is simply the difference between the 

 probable errors of the two tabular corrections applied, being, in fact, that 

 proportional part of the probable error of c corresponding to the fraction 

 of the vei-nier actually used. If the residuals indicate the existence of a 

 systematic error, a supplementary correction may be obtained by the 

 graphic process which has been described, though such an adjustment 

 will rarely be required, except when valuable observations have been 

 made with a sextant which had previously received some injury. If the 

 index-bar were bent, for instance, so that the two ends of the vernier are 

 unequally distant from the axis, it will be found that the divisions are 

 longest at the nearer end, and shortest at the more distant one. The 

 vernier corrections, like those pertaining to the limb, may be determined 

 with increased accuracy by several series of comparisons, preferably made 

 with different parts of the circle, and combined by methods too obvious 

 to require special explanation. 



For any given sextant reading the argument of the correction is not 

 the reading itself, but that of the point where a line of the vernier coin- 

 cides with one of the limb*. The readings 8° 59' 50" and 9° 0' 10" differ 

 only twenty seconds, but upon a sextant divided to 10' and reading to 10" 

 they refer to positions on the limb nearly ten degrees apart. This cir- 

 cumstance, which is not invariably mentioned in the text-books, is also of 

 considerable importance in determining eccentric corrections by the 

 methods commonly recommended. When extreme precision is desired, 

 the accuracy of an observation already made may sometimes be increased 

 by a device applicable to any sextant, whether its errors have been inves- 

 tigated or not. Find the point of coincidence of the recorded reading, 

 and, after setting the zero of the vernier exactly upon that position, read 

 the vernier at the other end ; then, setting the terminal line of the ver- 

 nier at the point previously occupied by the initial line, read at the zero 

 end. Subtract the nominal length of the vernier from the sum of the 

 two vernier readings and divide the remainder by three ; the quotient is 

 a correction to be applied to the original sextant reading. An error in 



* Every sextant reading is the sum of the limb reading and the vernier reading, 

 and may be readily separated into these two parts when the scheme of graduation 

 is known. If the vernier is of the direct or " short " form, almost universally 

 applied to sextants, the point of coincidence may be found by the following rule : 

 To the limb reading add the vernier reading multiplied by the number of divisions 

 in the vernier. Thus on the arc of a sextant divided to 10'', and reading to 10'''', 

 and which, therefore, has a vernier of 60 divisions, the reading 6° 49'' 50''^ is made 

 at 6° 40' +9^ 50^' X 60 = 16° 30'. For a reversed or " long " vernier, the same 

 product is to be subtracted from the limb reading. 



