15 
3. Euler (Mechanics) found different examples of these 
duplex solutions, and even gave rules for discovering them 
in certain cases. Some years after (in his Memoir of 1756) 
he expressly occupied himself with the subject (ib. 282). 
He was content to exhibit this duplicity of solutions as a 
paradox ; inasmuch as the equation which contains an 
arbitrary constant, and which we ought therefore to regard 
as the complete integral, does not include the other finite 
equation. This last is, however, equally a solution of the 
differential equation, contrary, as it seemed, to the principles 
of the calculus (ib. 287). 
4. It remained to discover the connection between these 
singular solutions and the complete integrals, as well as 
between the curves represented by the one and the other, 
and to refer the whole theory of these different solutions to 
the first principles of the calculus. This was done by 
Lagrange (ib. 290; 178). 
5. It is the characteristic of singular solutions that they 
appertain to the curves formed by the continuous intersec- 
tion of the curves represented by the complete primitive, 
when the arbitrary constant, a particular value of which 
distinguishes a particular primitive, is made to vary 
continuously (ib. 268, 269). 
Brisbane, Queensland, Australia, 
September 7, 1877. 
On the Formation of Hailstones, Raindrops, and Snow- 
flakes, by Prof. 0. Reynolds, F.RS. 
My present communication forms a continuation of the 
paper I read before this Society on the 31st of October, 
1876, “On the manner in which Raindrops and Hailstones 
are formed.’ 5 
To the contents of this paper I shall have to refer con- 
tinually — hence in order to render what I have to say 
intelligible it may be well for me to recapitulate some of 
the leading points in my former paper. The chief purpose of 
