360 Mr. Powell* i Appendix to M» Ramond's Instructions [Nov* 



extremes of the elevation. If we call the force of gravity at the 

 surface g, and that at the height z, g,,, it is obvious that 



7i must be reduced in the ratio of — , or we must take locr. 

 (h) "*" ^^S* \ /• "^^^^ ^^^^ term, from considering the law of 

 distance, may be converted into 2 log. (l + -) : [a being the 



mean radius of the earth, which may be substituted as very 

 nearly the distance of the lower station from the centre ; and 



instead of — putting f^-~,\ 



This last expression may be made use of in applying the 

 correction for gravity to the coefficient C. It will be sufficiently 

 accurate to take the expression for the force of gravity at tho 



mean elevation, which will be gi ^. And dividing by «% 



and neglecting all powers above the first, this becomes nearly 

 g, .-- . Taking also, instead of the indeterminate t, the 



mean temperature of the air at the two stations, we shall have 



p __ A (I - -0009028) g. 



These expressions for the diminution of gravity are deduced 

 by M. Biot through a series of analytical forms, in which he 

 traces the effects of the force in question from one stratum of air 

 to another, and then effects a summation. This is in accordance 

 with the elegant method he has adopted throughout the whole 

 investigation. In giving this outline, I have merely attempted 

 to state in general terms the grounds upon which each correc- 

 tional expression may be deduced; but for the details of the 

 analysis, the student is referred to M. Biot's tract. 



Fourthly, the formula now stands thus, 



^ = A(.-~8)„ (^ + -0028371 . C0S.2 +) (l + '-J^) 

 (log. (i) + 2 1og.(l + :-) ) (1 + i). 



Then developing 2 log. (l + -V and keeping to the first 

 power (since z is very small in respect to a), it becomes 

 -q--. Then multiplying the last two factors (keeping to first 



' power), we get log. (g) + ^ (log. g + ^) J whence (since ^^ 



