12 



SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 1 39 



when overriding harmonics are computed and removed. From the 

 examples given (out of about 10,000 cases available in my files) the 

 following 10 exact harmonics of 273 months are exposed as follows : 



1/4, 1/7, 1/8, I/I2, I/14, I/2I, 1/28, 1/35, 1/49, I/56. 



I 5 9 13 17 Q 15 29 J3 37 



Fig. 6. — Helena, Mont. Thirty-nine-month period in precipitation as cleared of over- 

 riding subordinate integrally related periods. Original tabulation, a; cleared curve, 

 b, with smoothed curve above. Note approximate sine form. Range, 27 percent of nor- 

 mal precipitation. 



While most removals of harmonic riders are done to clear periods 

 exceeding 1 5-1/6 months, many curves representing periods between 

 9-1/10 and 15-1/6 months required removal of harmonics of 1/2 or 

 1/3 of their length. An algebraic theorem affords a check on mistakes 

 of computation when clearing half periods. 



Let a periodic curve be represented by equally spaced ordinates 

 a, b, c . . . k, 1, m, and proceeding further, n, o, p . . . x, y, z. 



