1892.] facilitating the Reduction of Tidal Observations. 369 



Let abscissae (fig. 1) measured from along A'OA represent 

 magnitude of a. 



Since a, lies between + -J, the limit of the figure is given by 

 OA = OA' = 1. 



If magnitude of error (i.e. m.s. special hour), measured in special 

 time, be represented by ordinates, a line BOB' at 45 to AOA' re- 

 presents all the errors which can arise in the incidence of the m.s. 

 12 h in the schedule of special time. 



If a line W be drawn parallel to and above BB' by a distance ft, 

 we have a representation of all the errors of incidence of the m.s. ll h . 

 If a series of equidistant parallel lines be drawn above and below 

 BB' until there are 12 above and 11 below, then the errors of all the 

 m.s. hours are represented, the top one showing the errors of the 

 m.s. O h and the bottom one the errors of the m.s. 23 h . 



Any special hour corresponds with equal frequency with each solar 

 hour, and hence each mode of error occurs with equal frequency. 



It is now necessary to consider in how many ways an error of 

 given magnitude can occur. If in the figure AM represents an error 

 of given magnitude, then wherever MN cuts a diagonal line, it shows 

 that an error may arise in one way. 



It is thus clear that there are no + errors greater than J + 12 ft, 

 and no errors greater than f + 11 ft, and 



Errors of magnitude. 



J + 12 ft to -J + ll ft may arise in 1 way. 



i+ll/8 to + 10/3 2 ways. 



i + 9ft Sways. 



^ 10ft to |ll/3 23 ways. 



1-11/3 to -(1-12/3) 24 ways. 



-(1-12/3) to -(i-n/3) 23 ways. 



-(i + 9/3) to -(J + 10/3) 2 ways. 



to -(i + n/3) 1 way. 



The frequency of error is represented graphically in fig. 2. The 

 slope of the two staircases is drawn at 45, but any other slope would 

 have done equally well. 



A frequency curve of this form is not very convenient, and, as 

 there are many steps in the ascending and descending slopes, I sub- 

 stitute the frequency curve shown in fig. 3. This is clearly equivalent 

 to the former one. In fig. 3 all the times shown in fig. 2 are con- 

 verted to angle at 15 to the hour ; e accordingly denotes 15 ft. 



