of Variable Quantities. 11 



There are eight terms of the first kmd, such as [^' + 1, ?/ — 1, 

 ^+1] ; twelve of the second, such as [.r+l, ?/, ^ — 1]; six of 

 the third, such as [x-\- I, ij, ^] ; and only one of the last, viz. 

 [x,^j,t].^ 



Application to Ocean Meteorology, — This last process ap- 

 pears to nie to be applicable here. Meteorologists have divided 

 the ocean up into squares of 5° of latitude and 5° of longitude. 

 The logs of ships sailing over those squares are consulted for 

 meteorological observations ; and the results are classified by 

 months and squares ; and the mean result for any one element, 

 such as the. height of the barometer, is taken to be the average 

 for the middle of that square and .the middle of that month*. 

 Now it seems as though this were a case where smoothing is 

 justifiable, and that it is allowable to make the result for each 

 month depend in some degree on its neighbours both in space 

 and time. There are three independent variables, viz. latitude, 

 longitude, and time ; and in the previous formula we may 

 take these to be represented by x, y, and t respectively. 



Suppose, for example, we want to modify the mean height 

 of the barometer for any square e for, say, February, both 

 W'ith reference to surrounding squares and to the heights for 

 January and March, Then the rule for finding the amended 

 height is ; — Take the sum of the heights 

 for a, c, g, k for January and for March + ■ 

 tv/ice the sum of the heights for h, d,h,f 

 for January and March + twice the sum ■ 

 of the heights for a, g, c, k for Febru- 

 ary + four times the sum of the heights ■ 

 of 6, d, A, / for February + four times 

 the sum of the heights for e for January ■ 

 and March + eight times the height for 

 e for February, and divide the result by 64. 



It must be observed that it is not necessary that the smooth- 

 ing should be carried to the same extent for x, y, and t. If, 

 for example, we wish to smooth only once for time, the for- 

 mula will be different, and the result will be applicable to the 

 beginnings of the months instead of the middles. A knowledge 

 of the particular requirements of the case is the only guide to 

 the amount of smoothing which is expedient ; but the formulas 

 are so easy to construct that it does not seem worth while to 

 give any other forms. 



In concluding this part of the subject, I may mention that 

 the proposed processes may be extended so as to allow various 

 weights to the various observations. 



'Terrestrial Meteorology. — There are a number of observa- 

 * I owe this explanation to Mr. Francis Galton. 



d 



f 



