Tem/perature of Mercury in a Siphon Barometer. 261 



Frustum dd"=^ (&/-*) [r' + [r+(6/-6) sin. ^']2+r[r-l- 



{h,-h)sm.6'Y] (20.) 



It is evident, whatever may be the forms of the tubes, that 

 capacity of D'D''= capacity of d'd" {^\.) 

 But,- from the figure cap. D'D''=cap. DD^' — cap. DD' 

 cap. <^'o?'''=cap. dd"-\-Q,^'^. dd'. 



Equating the second members according to (21,) and transpos- 

 ing, we have 



cap. DD''— cap. dd"=Q,d,i^. DD'+cap. dd' ; 

 or, denoting the sum of the second members of (17) and (18) by 



(10 (22) 

 the second member of (19) by (K,) (23) 

 and the second member of (20) by (L,) (24) 

 wehaveK-L=I(25.) 



In hke manner, if the observations {a,,^ b,^, i,„) {a^, 63, ^3,) 

 (a^, 64, t^,) be compared respectively as above with (a, b, t,) we 

 shall have K'-L'=F, K"-W=^V', K'''-L'"=:F'; (26) the 

 terms of these equations being functions similar to those of (25.) 



The four equations (25) and (26,) after correcting the readings, 



are sufficient to determine the unknown quantities — j &, 6' and 



( -rP-^V'i which these equations contain. 



From the minuteness of the angles (<9,) {d\) we may use (<?,) 

 (6') in the places of sin. ^, sin. Q' ; and neglect the terms which 

 contain the second powers of these angles. This will materially 

 abridge labor without impairing the practical accuracy. 



As our limits will not permit us to discuss the general question, 

 we will select that particular case only in which the two branch- 

 es are cylinders of unequal diameters ; and which is the most 

 important, if not the only one that needs to be regarded in the use 

 of the barometer. 



Here 5^0, ^'=0 



K=5TR2(^^_gjj according to (23) 

 ■L^nr^{b-b) " (24) 



l=ne(K^pJ\.r^p') " (22) 



Substituting these in (25,) and dividing by {-rrr^^) we have 



%{a-a)-{b,-b)^e^^p+p) {t'-t,) (27.) 

 Vol. XL, No. 2.— Jan.-March, 1841. 34 



