Capillary Surface inside a Tube of Small Radius. 133 



that equation (xiv.), obt lined on the assumption that the 

 liquid surface is hemispherical, is precisely the equation 

 which does, to the first order, correct for deviations from 

 sphericity, and already embodies within itself the fact that 

 the drop is spheroidal in outline. The later refinement of 

 treating the drop as spheroidal and arguing by the "weight" 

 method, merely introduces a term of higher order into 

 (xiv.) or (xvi.), 



§ 5. It is worth noting that direct measurements of li x and 

 r Y afford a feasible method for the determination of contact- 

 angles. M. Sentis' equation (xiv.) gives a value for a 2 which 

 is quite independent of the angle of contact. Also from 

 (vii.) we have 



1 cos i r Y . f 2 sin 2 z cos £—1 1 , .. . 



77- = f- ,r^ sec l i *+ -7 2~ — r • ( xvl1 -) 



Y\ x i\ la 2 I 6 cos 2 i J 7 



Substituting in (xiv. a) the values of — and rr- from (xvii.) 



ill K 2 



and (xi.) respectively, we have 



hi cos i , i\ . f. 2 sin 2 £ cos t— 1 1 1 i\ 



2a* r, '2a* I ' 3 cos 2 i J T r Q ' 6a 2 ' 



cosz=--V- 



2a- r 2 



Approximately ._ / 7^ 



COS 2 — ?'i I ^ 



f i 2 sin 2 z cos 2 — 11 1 ?' 2 



1 1 + o • r — + - + 7T- 



I 6 cos"* 2 J r 2 ba' 



or .... (xviii.) 



ryr 2 r? J - 2 sin 2 i cos 2—11 



— ^-r — o- 2 1 1+ 9 2^ — r sec *- 



ba- zuS \. 6 cos- 2 J 



.... (xix.) 



._ (K _L_-!1\ 



~ 7l \2a 2 r 2 6a 3 /' 



which value of 2, substituted in the small term of (xix.), 

 gives a still closer approximation to cos i. 

 If 2 = 0, (xix.) becomes 



6a 2 (r 1 +r 3 )=r ] r 2 (3/i 1 — r ± — r 2 ), . . . (xx.) 



and a convenient test of a zero contact-angle consists in 

 determining whether the value of a 2 calculated from (xx.) 

 (which depends on the assumption that the contact-angle is 

 zero) is in agreement with the value calculated from (xiv.), 

 which is independent of this assumption. If the difference 

 between the two results lies outside the limits of experi- 

 mental error, equation (xix.) can be used to calculate the 

 value of 2. 



§ 6. Before proceeding farther, it may be well to empha- 

 size the limits within which these approximations can safely 

 be applied. Two assumptions have been made in order to 



