the Depression of the Mercury in the Tubes of Barometers. 5 



-£- ; which being done, and the value of -— - substituted, we 

 shall get this equation in partial fluxions, viz. 



a . da da z a da 4 , * 



dz dz t t ' dt y^/l—z* * ' 



to which we must join the value of ~^- 9 viz. 

 I next write the equation (1) in this manner, 



1 d . a z3 da t d.alz* 4 



t z*dz dt 2zdz .V i_ JK 2 



and assume, co = § + Az 2 + Bz 4 + &c. 



§, A, B, &c. being functions of t. Substitute this value of co, 

 and expand the radical on the right-hand side of the equa- 

 tion ; then, by equating the coefficients of the like powers of 

 z, we shall get, 



&c. 

 These equations give us, 



* = *-4 +&c. 



A = T - T +&c. 



&c. 



There is another consequence of the first of the equations (3) 

 which it is necessary to notice. Put 



then we readily derive from the equation mentioned, 

 — i 3 dx -i,.y 



and by integrating in a series, without introducing any arbi- 

 trary constants, which the present purpose does not require, 

 we shall get, 



' Oi *<* a "I" 2». 32. 4a. 5 "*" ^ C * 



If 



