394 Royal Society. 



from the z line. Then the numerical part of the general term will 

 be 



n(\, + M, + Qn(x a +)u.+r i )n(\ ) ,+/ lt! ,+v <l ) 



nx,nju,[i>' 1 nx 2 n^ 2 iir 2 n/\ 3 n j u ;J ii>' 3 



n(L+/»)n(M+f)n(N+r) 



,' n'/,n'?« 2 n'M.,n : /.,n-TO.,n 2 «, 



YVlJ.Vm l \VnJ\njVm x Wn, "j 



[n i / s n , jn s ii i » s n 4 / 3 Li a »i 3 n9» s J 



p 



x no,ntf 2 . . .no«n^,n^ 2 . . n^n^m/v .. n^ y ' 



where in general Urn means 1.2.3...m : as regards D, it is the fol- 

 lowing determinant, viz. 



^1 + /*i + '' i v v Lj M 3 N 3 



v x 2 + ^ 2 -f-»' 2 v L 2 M s N 2 



v v \ 3 +p 3 +Vs L, M, N, 



Xi X 2 X 3 Li + L 2 L 5 +p v v 



/i, fit fi 3 v M, + M a + M, 



v, v 2 v 3 v v N, + N 2 + N 3 + r 



The result, for greater brevity, has been set out in the above pages 

 for the case of •&, a function of three variables, but the reader can 

 have no difficulty in extending the statement to any number. In 

 the case of a single variable, the formula can easily be identified 

 with that given by Burman's law. It is noticeable that the deter- 

 minant above written is of the form 



Apqr + Bpq + Cqr + Dqrp + Epe + F q + Gr, 



the part independent of p, q, r being easily seen to vanish. More- 

 over, A, B, C, D, E, F, G, H are all essentially positive, so that D 

 can only vanish (except for p=0, r=0, <?=0) by virtue of one con- 

 dition at least more than the number of the variables. 



Feb. 15, 1855.— Thomas Bell, Esq., V.P., in the Chair. 

 The following communications were read : — 



" On the Computation of the Effect of the Attraction of Mountain- 

 masses, as disturbing the apparent astronomical latitude of stations 

 in Geodetic Surveys." By George B. Airy, Esq., F.R.S., Astro- 

 nomer Royal. 



The author commences with remarking that his surprise had been 

 excited by the result obtained by Archdeacon Pratt*, namely, that 

 the computed attraction of the elevated country north-east of India 

 considerably exceeds the disturbance which it was sought to explain. 

 But on consideration the author perceived that this result might have 

 been anticipated, on the extensively received supposition that the 



* Proceedings of the Royal Society, December 7, 1854. 



