650 REPORT— 1900. 



' Stirling's first ' formula can be written U:g^ — u„ 



= {.V + (AS)-> X . a5 + (AS)-- .V . (Adf + &c.} ^^Mo 



+ ^±^ {(A8)-^ X . AS + (AS)-- X . (AS) + &c.}mo ; 



and ' Stirling's second,' with origin midway between m,, and Mj, can be written 



u^= {1 +(A8)-' 1 .AS + (AS)-- 1 . (AS)2 + &c.}^f^i 



+ ^l±il'(l + (AS)-' 1 . AS + (AS)-- 1 . (AS)'^ + &c.} Am„ ; 



but here, instead of the inyerse functions vanishing with x, A"^ is to vanish^ for 

 x=h and S-' for x= —h. 



12. On Newton^ s Contributions to Central Difference Interpolation. 

 By Professor J. D. Everett, F.R.S} 



After a portion of the preceding abstract had been sent to the printer, I 

 discovered that the two formulae known as Stirling's are really due to Newton. 

 They are given in Prop. 3 of Newton's ' Methodus Dift'erentialis,' which is the closing 

 one of a collection of minor treatises by Newton, published, with his permission, in 

 1711 by W. Jones, who states in the preface that this particular tract has been 

 transcribed by him from a manuscript in Newton's handwriting. 



I have also found in Lemma 5, which follows Prop. 40 of the third book of the 

 ' Principia,' what is doubtless the earliest statement of the ordinary interpolation 

 formula, now usually written 





Newton gives it in a geometric shape, x, x—\, x — 2, &c., being represented by 

 lines, and he defines Au^, not as «, — u, but as u„ — u^, with a similar convention 

 for higher difierences, thus reversing the signs of all odd differences as compared 

 with modern notation. Owing, 1 presume, to these disguises, the formula appears 

 to have hitherto escaped recognition. 



Department II. — Meteorology. 

 1. Report on Meteorological Photography. — See Reports, p. 56. 



2. Report on Seismological Observations. — See Reports, p. 59. 



3. Fifth Report on the Use of Kites to obtain Meteorological Observa- 

 tions at Blue Hill Observatory, Massachusetts, U.S.A. By A. Lawrence 

 RoTCH, S.B., M.A., Director. 



Satisfactory progress has been made in the work since the report presented at 

 the Dover meeting, and it is gratifying to observe that the method of exploring the 

 air by means of instruments, recording graphically and lifted by kites, which was 

 initiated at Blue Hill in 1894, is now being extensively used on the continent of 

 Europe. 



' Published in extenso in the Journal of the Institute of Acttiaries, 1901. 



