594 Proceedings of the Royal Society 
shell from the mouth of a funnel ; the mode in which two bubbles 
unite ; and the process of cutting one into two or more. 
A statical investigation of the form of an unclosed film, blown 
with coal gas, was given (the kinetic problem presenting very 
grave difficulties), and the results were shown to be in accordance 
with observation, so far as the eye can follow the rapid change 
which takes place in the neck of the film just before the closed 
bubble is detached. 
Professor Tait called attention to the exquisite manner in which 
the molecular motions in the film may be exhibited by employing 
the posterior surface of a large bubble as a concave mirror to form 
a small bright point from a beam of parallel rays, and receiving on 
a screen the light diverging from this point after it has passed 
through portions of the anterior surface. 
He also noticed that the spectrum of the reflected light shows 
very effectively the phenomena of interference, supposed by Yon 
Wrede to account for the dark lines in the solar spectrum. 
2. On Functions with Recurring Derivatives. By Edward 
Sang, Esq. 
In a previous paper, it was pointed out that the characteristic 
problem of the third branch of the higher calculus, is to discover 
the relation between the primary variable and its function, when 
the relation subsisting between the function and its derivative is 
known. The present paper treats of the solution of the simplest 
case of this general problem, that in which the function is equal or 
proportional to its derivative. 
The proposition in hand is naturally divided into cases, accord- 
ing to the order of derivation : The first two of these can, by well- 
known artifices, be brought under the dominion of the integral 
calculus, and their relations can therefore present nothing new. 
But for the sake of the continuity of the treatment, and of certain 
relationships which otherwise could not have been so well explained, 
they have been discussed in the paper. When we inquire into the 
nature of the function which is equal to its own first derivative, we 
arrive at the exponential function, and at the basis of Neperian 
Logarithms of this function e^, the development is 
