CORRECTION OF THE HEIGHT OF THE BAROMETER. §7 



the radius is *5, and to find the mean radius of the arc of 

 30°, we have 2 s ('13 — |*) zz *6l, ss — -26 s zz — 01 

 s — 13 — ■ \/'00b9 — -0463 : but the versed sine of the arc 

 of which this is the sine, in the circle of curvature at the ver- 

 tical point, is '01552, which is to be diminished in the ratio 

 of -13 — *0J54 to *13, and becomes "01371 ; and deducting 

 tjiis from -5, we have *48b'3 for the value of x\ when n zz 

 •008GG. Hence we find /; zz '0948, a zz *0038, and the 

 marginal elevation '151, which is so near the assumed value, 

 that no further correction is required. In the same manner, 

 for tubes of §• and f inch in diameter, we find a zz *0374, 

 and *130, and the marginal elevation -]62 and "220 respec- 

 tively. But it would be rather more accurate to compute 

 the extent of a portion of the curve, somewhat greater than 

 60°, by means of the series. 



We may also obtain a series, in a manner nearly similar, Series hi terms 

 for determining the relation of the arc to the absciss and ° tiearc * 

 the ordinate ; and such a series must represent the proper- 

 ties of the curve in a more general manner, and may, in 

 some cases, be move convenient for calculation, at the 

 same time that it affords a mode of verifying the results 

 •which we have already obtained. Taking the expres- 

 sion fx y x \S {x 7, + j a ) zzmxy, we may put x z -\-jf z zz « a , 

 x zz z + A z 3 + B z s + C z 1 + . . . , and y — a + b z* + 



x z y a X* 



cz+ + dz* + . . . ; then -tt + 4* = I ; but — = 1+6 



z, z z 



Az z + (10 B + 9 A*) z 4 + (14 C + 30 A B) z 6 f (18 D 



+ 42 A C + 25 B*) z % + . . . , and 4 = 4 i 2 s a + 16 b 



z 



cz* + (24 b d -f- 16 c a ) z 6 + (32 b e + 48 c d) z s + . . . ; 



whence, by comparing the homologous terms, A zz — }i*, 



~l64c-9 A p — 24 bd — l6cc- 30AB 



x> zz ' , C — . 



10 14 ' 



.' —3-2 be — 48 c d — 42 A C — 25 B B 



and D zz . Again, 



18 5 



for the fluent fx y x, we have x y zz a z -\- (a. A + b) z % 



+ (aB + a+c): s + ..,, xzzz+3Az z z + 5B 



s 4 k -\- . . . , and fx y k zz a — -f- (a A + H 3 A a) 



