IN RAIN AND SNOW, IN THE UNITED STATES. 153 



To free the ratios of the preceding table from the accidental irregularities, and 

 to exhibit the nature of the fluctuations from year to year more distinctly, we unite 

 them into groups formed of stations where the annual rain-fall appears subject to 

 the same laws. The number and extent of these groups may be recognized by 

 means of the investigation of the annual distribution as shown by the type curves, 

 and the proper selection of stations so combined may be verified, to some extent, 

 by the agreement within the accidental irregularities of the ratios themselves. 

 These means, the number of stations from which they were derived, and the fourth 

 order of successive means are given in the following table for a number of groups 

 or types. The successive means 1 are introduced to smooth down the incidental 

 irregularities of the phenomenon as a means for the further study of the annual 

 fluctuation. 



Group I is composed of stations on the Atlantic sea-board from Maine to Vir- 

 ginia (see Diagram 3) ; Group II covers the State of New York, it includes also 

 Hanover, Burlington, Amherst, Montreal, and Toronto; Group III is formed by 

 Forts Eipley and Snelling, Dubuque, Milwaukee, Muscatine, Fort Madison, and 

 Ottawa; Group IV covers the Ohio Valley from Pittsburg westward, and includes 

 Eastern Missouri; Group V is formed by Forts Gibson, Washita, Towson, Smith, 

 and Washington, Ark.; Group VII includes New Orleans, Baton Rouge, Mount 

 Vernon Arsenal, Huntsville, Greensboro', and Fort Barrancas; Group VIII com- 

 prises the Southern coast from Virginia to Florida; and Group IX the Pacific 

 coast stations San Diego, San Francisco, Sacramento, Benicia Barracks, and Fort 

 Humboldt. 



Results from Groups I, II, IV are tolerably trustworthy, but the remaining ones 

 can only be taken for rough approximations on account of an insufficiency of stations. 



The decimal points are omitted in the tabular quantities; they are expressed, 

 therefore, in percentage of the mean amount. 



1 The following formulae may also be employed, occasionally, to effect the same ; they involve no 

 special supposition as to curvature (being based upon the consideration of straight lines), and are 

 directly derived from the average ordinate y m = ^ (j/j-f- y 2 ) ; 

 for three ordinates 2/m = ? (y, + %!/ 2 + 2/ 3 ) > 



f ° r f our " 2/m = i (</, + 3y 2 + By, +y t ) ; 



for fi ™ " y » = T V (y, + il/,+ 6y 3 + iy t + y t ) ; 



for si * y» = A (2/i + 5 2/» + 10y, + lOy, + by, + y 8 ) ; 



for seven " y a = & (y, -f 6y 2 + 15y, + 20y 4 + 15y, + 6y 6 + y,) ; and so on ; 



for n+1 ordinates the coefficients are those of the n" 1 power of a binomial and corresponding to suc- 

 cessive means of the n th order. 



At what order of successive means we should stop depends entirely on the nature of the irregulari- 

 ties ; care must be taken not to eliminate by the process any of the characteristic features. 



20 July, 1871. 



