2a 1 = rh 



i + 



Manchester Memoirs, Vol. Ixv. (1921), No. $ 11 



Approximately, 



2a 2 = rh + \r 2 , 



and this value, substituted in the small term on the R.H.S. of (v) gives> 

 after a few simple reductions 



\ L h ~ W {2 log < 2 - x) + w {2 loge 2 " r) l 



= rh(i + j~ - 0-1288^ + °'°429^) .... (vi) 



Now consider the results obtained by more complex methods of analysis. 

 Poisson 1 gives, in effect 



/ 1 r r 2 \ 



2a 2 = rhyi H — -= - o'1288-pj. 



By a very much longer analysis, Rayleigh 2 has obtained a third-order 

 correction, his value being 



2a 2 = rh[i + .|£ - 0-1288^ + o-i3 12 ^) • • (vii) 



and it will be seen that equation (vi), which only professes, to be accurate 



. r 2 . 

 as far as the term in j~ inclusive, differs very little numerically from 



Rayleigh's equation. 



An interesting and simple way of approximating to the correction terms 

 was first given, apparently by Desains, who treated the meridional curve 

 as elliptical. One way of arriving at the correction terms on this assump- 

 tion is given by Mathieu 3 and by Rayleigh, 4 but it can be exhibited in a 

 much more simple and direct way ^s follows : — 



Consider the outline in Fig. 1 {p and c) as the outline of a semi-ellipse 

 of semi-axes rand^. Then, equating 27rrT to the weight of liquid raised 

 (including the weight of the liquid in the meniscus) we have, 



27rrT = 7rr 2 hpg+ \"nr 2 bpg 

 or, 2a 2 — rh + \rb . . ... . (viii) 



But, where R is the radius of curvature at the origin O, we have accurately 



2a 2 = Rh, 



and R, the radius of curvature at the end of the semi-axis minor of an 



r 2 R 

 ellipse is equal to -y-. 5 Hence 



r 2 r 2 h 



2a 2 = — h or b = — h . 

 b 2a 1 



which value of b, substituted in (viii) gives 



I2# 4 — 6rha 2 — r % h = o. 



1 " Nouvelle Theorie ..." (1831), p. 112. 2 Rayleigh, I.e., p. 189. 



3 Mathieu, I.e., p. 49. 4 Rayleigh, I.e., p. 190. 



5 See any text-book on the differential calculus. 



