HTDTiAFLIC INVESTIGATIONS. 117 



-— , gf-— 2 h f so tliat h is always equal to half the difference 



_- x 



between g and the actual height of the column above the 

 given point, or to half the height of the point above the base 

 of the column. 



If two values of x, with their corresponding heights, are 

 given, as b and x, corresponding to c and d, and it is rer 



quired to find a; we have — - — : c: : — '■ — : d, dbx — dax 



b x 



, , . dbx — cb x b d x — ch 



zcox-r-coa, and a 32 — -— or — ~ -: . 



a X-—C ft a a x -rr- c x 



Thus if the height equivalent to the tension vary in the 



ratio of any power m of the diameter, so that, n being a 



small quantity, x m b (l + n) and d zz c (I + mn\ 9 — Sf 



ftc((l-J-n).(l + «u)-l) mn + n 



' —rr _____^_ since the 



be ((l-f-w). (1 -f mn--(l + w) " ■*• 



r. • , b w+1 r, 

 square ot n is evanescent, and — zs . ror example, 



b 5 



if m s 4, — 35 — » and if tw — 2, & : a : : 3 : 2. 

 a 4 



IV. Of the Magnitude of a diverging Pulsation at different 



Point?. 



_ 



The demonstrations of Euler, Lagrange, and Bernoulli, Magnitude of 

 respecting the propagation of sound, have determined, that a diverging 

 the velocity of the actual motion of the individual particles different 1 ** 

 of an elastic fluid, when an impulse is transmitted through points. 

 a conical pipe, or diyerges spherically from a centre, varies 

 in the simple inverse ratjo of the distance from the yertex or 

 centre, or in the inverse subduplicate ratio of the number 

 of particles affected, as might naturally be inferred from 

 the general law of the preservation of the ascending force 

 or impetus, in all cases of the communication of motion be- 

 tween .elastic bodies, or the particles of fluids of any kind. 

 There is also another way of considering the subject, by Waves, 

 which a similar conclusion may be formed respecting waves 

 diverging from or converging to a centre. Suppose a straight 



wave 



