194 DISCUSSION OF THE ANNUAL FLUCTUATION 



at Marietta and Providence between October 27, and November 2, is contradicted 

 by the ordinary fall of temperature observed at Salem during this period, but appears 

 supported by Toronto. 



The smooth curves, given in the Marietta and Providence diagrams, which cut 

 off the zigzags, equally, above and below, are obtained by the method of successive 

 means, and in this instance represent the sixth order of means.' This process 

 facilitates comparison and enables us to construct tables of daily temperature, the 

 values of which have thus become more consistent by the removal of the greater 

 accidental irregularities. 



In the tables which follow, the annual fluctuation is given either directly by the 

 daily ordinates or by those of smooth curves, obtained by tlie process just explained, 

 or by means of Bessel's periodic function with constants supplied by observation, as 

 stated at the top of each table. 



The director of the Toronto observatory noticed the curious fact, that the daily 

 means or normals of temperature made out by General Sabine for the epoch 1841 

 to 1852 had now become totally inapplicable, in consequence of wliich a new set 

 of normals was prepared, employing the series of observations from 1859 to 1868, 

 and calculating the table with the help of Bessel's periodic function as had been 

 done before. 



The two sets of tables given for Toronto will, therefore, represent the variability 

 of the annual fluctuation for two epochs not very remote from those when the 

 extreme values obtain, as has been found from a further study of this phenomenon 

 of the shifting of the epoch of maximum cold and of apparent changes in the 

 curve of the annual fluctuation.^ 



On account of this variability of the annual fluctuation, the years of observation 

 from which the daily means were deduced, are stated at the head of each table. 



' Supposing 2/i y., y^ i/^ y^ y^ y-, to represent consecutive values of the daily temperature, the resulting 

 mean of the sixth order and corresponding in point of time to the middle ordinate y^ will be given by 



A {Vi + %. + 152/3 + 201/, + 152/, + 62/e + 2/,} 

 and in general for n ~\- 1 ordinates, the co-efiicients are those of the nth power of a binomial and the 

 divisor equals their sum. 



No precise rule can be given prescribing the limiting number of successive means, but as the 

 values converge towards a constant, at first rapidly and afterwards more slowly, it will soon be found 

 that after repeating the process a few times very little impression can be made on the results by 

 continuing it, which sufBciently indicates that we have arrived at a practical limit. We may either 

 compute directly by means of the formula, or we may set down each series of consecutive means; the 

 latter process offers the advantage of a partial check in the regularity of progression of the numbers 

 standing in the same horizontal line. It will also be convenient to stop at an order of an even 

 number, in which case the resulting means refer, in point of time, to noon, whereas odd numbers 

 (which may be written between the line) refer to midnight. 



^ Referring the reader to a subsequent part of this paper for the analyzation of the results connected 

 with this inequality, it may be stated that it probably exists over the greater part of the United 

 States east of the Mississippi River, and, perhaps with some modification, also in other parts of the 

 country; allied with it, but not necessarily connected, there appears also an inequality in the amount 

 of greatest cold and heat extending over a number of years, which, however, leaves the annual range 

 almost undisturbed. These inequalities are necessarily of a periodic nature, and consequently our 

 daily means, in order to become truly normals, must comprise at least one full period (or at least 

 half a period if the curve be regular and just includes the maximum and minimum). 



