12 Mr. G. H. Darwiii o/i Fallible Measures 



tories on the land at which observations are taken at the same 

 hours of the day all over the country ; from these results, maps 

 are drawn showing the form of the "isobars" for each day. 

 After the observations have been reduced to the sea-level and 

 corrected in other ways, they may be considered as correct, 

 and the isobars give a graphical illustration of the successive 

 deformations of the barometric surface. Land meteorology 

 serves, then, to give quite a different kind of result from those 

 of ocean meteorology. In the latter the result is the mean 

 heights of the barometer at stated places and times. The 

 oceanic barometric surface, as far as we know it, is ideal, and 

 does not correspond with ita real form at any one time. In 

 oceanic meteorology the smoothing process seems justifiable ; 

 for we only seek t'o study the main features of the changes. 

 In land meteorology this is not the case; for we seek to dis- 

 cover the details of the changes. To return to the former 

 metaphor — in one case the law of the waves is sought, in the 

 other the law of the ripples. 



The observatories are scattered irregularly over the country; 

 and it seems probable that the results would be more useful 

 and more easily interpreted if they could be distributed at 

 regular intervals of space. They are already regularly distri- 

 buted as regards time. My present object is, then, to give a 

 formula (which is, as far as I am aware, new) for the reduc- 

 tion of observations scattered irregularly, to regular stations 

 equidistant in latitude and longitude. It is a problem in in- 

 terpolation of the ordinary kind where the ordinates are not 

 fallible. 



The problem is to find a continuous surface passing through 

 the tops of a number of irregularly spaced ordinates ; and it 

 may be solved by an extension of Lagrange's well-known 

 formula for interpolation in two dimensions. 



Let.'^Q, y^; Xi^yi', &c. ; o^m yn be the coordinates (latitude 

 and longitude) of a number of points, and let ^q, Zi, &c., Zn be 

 ordinates (barometric heights) corresponding to these points. 

 Lagrange's formula suggests the following as the equation to 

 a surface passing through the tops of 2-^, z^^ &c., ^^. 



^^^ {^ --%)(y -yi){x -X2){y-y^) ♦ . » {^ -Xn){y -yn) 



" K-'^l)(yo-^l)(^^0-'^2)(yo-y2) . . • 0^O-''''«)(yo-i/«) 



^^ (.a; -■^o)(y -yo)0^ -xi){y -y^ • " G^ -scn){y -yn ) . 

 ^ {x^—x^){yx—yo){xi—x2){yi-y2) • . • {xi—^^n){yi—yn) 



+ &C. 



^^ U^ -xo){y -yo)(^ -^i){y -yi) - • '(^^-^^^n-Q iy -yn-x) 



''lXn — ^o)(yn-yo)(^'>^n — ^'^l)(yn-yi)'--{^V^ — ^'l'n-l){yn—yn-iy 



