6 G. Barns — Spiral Goniometry i/n its 



certain of the circles on either side of the true angle are too 

 faint to be visible, therefore, the error of a double reading- 

 need be no larger than has been pointed out above. For 

 instance in a dial in which consecutive radial millimeters were 

 each marked by a partial circle of 60°, in such a way that a 

 millimeter corresponded to a degree of arc, the reading was 

 made by an index containing open sectors of 30°. The follow- 

 ing results were obtained : 



First reading, a 17 27 37 47 58 68. etc., millimeters. 



Second reading, « + 30° _ 51 62 72 82 92 10K etc., millimeters. 



Difference ...... 34 35 35 35 34 33.' etc., millimeters. 



K2a + 30)— 15° 19 - 0° 29 0° 39'5° 49-5° 60'0° 69-5°, etc. 



True angle 20° 30° 40° 50° 60" 70° 



In these cases the attempt was made to increase the angle in 

 successive steps of 10° each by special adjustment, as shown 

 in the last line. The angle found as the mean of the two 

 readings does not differ from the true angle by more than a 

 degree, part of which is referable to the preliminary adjust- 

 ment as well as to actual irregularities of graduation. In spite 

 of this good result it is seen that the differences (third row) of 

 corresponding readings (first and second row) is in excess of 

 the angle of the index (30°) by 3° to 5°. This shows, there- 

 fore, that when the system is spinning around, the projecting 

 ends of about two circles on each side of the circle correspond- 

 ing to the true angle, are so short as to escape detection. 



6. Multiple spirals. — Although the preceding paragraph 

 indicates that faintness of graduation does not necessarily 

 vitiate the method, it is desirable to have the marks come out 

 as early as possible. This end, together with all the sugges- 

 tions of the preceding paragraph may be promoted at once, 

 and even in a relatively small dial as follows : A smaller angu- 

 lar distance between the paired spirals is chosen and the 

 advantages of continuous double reading and ot repetition are 

 secured by employing more than two open sectors on the index. 

 In this case if the design is judiciously constructed as many 

 systems of paired spirals may be traced out on the dial as 

 there are open sectors on the index, the whole representing a 

 set of equidistant whorls. If therefore when both dial and 

 index are spinning around together, the protruding end of one 

 of the partial circles which just escapes detection in the above 

 designs is a, visibility has now been increased to the extent 

 that a./n just escapes detection, where n is the number of 

 sectors. 



A good arrangement of this kind is shown in figure 4, in 

 which the dial AA' contains three sets of paired spirals sub- 

 tending arcs of 60° each, with a blank space of 60° between 

 consecutive pairs. The index is shown at BB' and has three 



