of Edinburgh, Session 1879 - 80 . 
325 
for the equation to the surface, he deduces the approximate 
formulae, in which a = 
A 
2 _ge ah -e~ aJl 1 
a e ah - ah ' i _ a 2 a 2 (e uh - e~ ah ) 
2 
2 = 
a(e ah + e~ ah ) 
• a) 
• ( 2 ) 
the first of which gives the velocity of transmission, the second the 
height of the wave. 
In the latter part of the paper he applies his method to canals 
having a vertical section of any shape whatever, and deduces the 
following elegant formula — 
c 2 = g 
area of .vertical section 
breadth at surface 
for the velocity of propagation. This gives the result, for canals of 
triangular section, that the velocity of propagation is that in a 
rectangular canal of half the depth. This conclusion is tested by 
means of Scott-Russell’s observations, and is found to he in close 
agreement with fact. 
The same result was also arrived at independently by Green, who, 
in point of fact, anticipated Kelland in the matter, for he gives it in a 
note read before the Cambridge Philosophical Society on the 18th 
Pebruary 1839, whereas Kelland’s paper was read on the 1st April 
of the same year. Soott-Russell’s observations were the exciting 
cause of both investigations, which have little in common beyond 
this particular result. 
In a memoir on General Differentiation ( a Trans. R.S.E.” xiv.), 
read December 1839, Professor Kelland deals with one of the most 
abstruse and difficult branches of analysis. The process by which 
we extend the meaning of the symbol x ™ where m is integral to the 
case where m has any value whatever, is familiar enough, although 
it has its difficulties as every algebraist knows. General differen- 
tiation is a problem of a similar kind hut of a much higher order of 
difficulty. Thus, 
are symbols which have for their effect to deduce in a particular 
