Dec, 1915] Geometry of Translated Normal Curve 69 



lines, ^ = is the X — axis ; |S = 1 is a line parallel to the X — axis. 

 Hence we may safely assume that the variation from one col- 

 umn to the next and from one line to the next is linear for 

 values of /3. That is, ordinary first difference interpolation 

 methods are applicable. 



As regards the system of e curves we have for instance 

 € = .128 at (X = .050, /c = 0); again, at approximately (.045, .060) 

 and (.40, .085). We are therefore warranted in assuming the 

 applicability of first difference methods to interpolation between 

 the e curves. 



As an illustration let us find the values of X and k for e = 0.112 

 and j8 = 0.044. On inspection of the table it is seen that X lies 

 between 0.30 and .035 and k between .090 and .095. When 

 K = .090, X = .033 for e = .112. When k = .095, X = .031 for e = .112. 

 For ,3 = .042 and k- = .090, X = .033 and for /3 = .046 and k' = .095, 

 X = .031, € = .112 in each case. Hence, to first differences,. 

 X = .032 and k = .093 for € = . 1 12 and /3 = .044. For interpolation 

 in parts of the table showing more rapid variations appropriate 

 methods will suggest themselves. 



Taken geometrically the table represents two distinct sys- 

 tems of curves, with each curve of one system intersecting all 

 the curves of the other system. Therefore, a pair of values for 

 X and K can always be found for values of e and 8 within the 

 range of the table. 



Department of Mathematics, Ohio State University. 



