475 



ental functions under a radical. The radical must, however, not be 

 higher than the square root ; for, although it be true that if we take 

 the case of inserting two means between two quantities, the geometric 

 will still lie between the arithmetic and harmonic means, we have 

 nothing to show what the second step of approximation is to be. 



The third arithmetic mean is, in the case of elliptic integrals, 

 sufficiently near for working with seven figures. The resulting for- 

 mula, in the case of the elliptic integral of the third kind, is far from 

 being simple ; but it is practicable, and it requires none but the ordi- 

 nary trigonometric and logarithmic tables. This complexity is in 

 reality due to the extremely complex character of the function itself, 

 as is well known to everv one conversant with its transformations. 

 My method becomes sufficiently simple when applied to complete 

 elliptic functions of the first kind. 



My own opinion is that this method affords as easy an approxi- 

 mation as the nature of the elliptic and ultra elliptic integrals, at 

 least in their general form, admits, that it is simpler than the use of 

 Jacobi's functions or Y, and that except in isolated cases, there is 

 no advantage to be derived from the computation of tables of such 

 auxiliary functions, so far as the mere computation of elliptic functions 

 is concerned. 



II. " On the Lunar Diurnal Variation of Magnetic Decimation 

 at the Magnetic Equator." By JOHN ALLAN BROUN, 

 F.R.S., Director of the Trevandrum Observatory. Re- 

 ceived March 28, 1860. 



This variation, first obtained by M. Kreil, next by myself, and 

 afterwards by General Sabine, presents several anomalies which re- 

 quire careful consideration, and especially a careful examination of 

 the methods employed to obtain the results. The law obtained seems 

 to vary from place to place even in the same hemisphere and in the 

 same latitude, and this to such an extent, that, for example, when the 

 moon is on the inferior meridian at Toronto it produces a minimum 

 of westerly declination ; while for the moon on the inferior meridian 

 of Prague and Makerstoun in Scotland it produces a maximum of 

 westerly declination. No two places have as yet given exactly the 



