58 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Cayley, by considering two different forms of development — Bezout’s of 
1764 and his own of 1847 — arrives at the equations 
*(m, n) = m$(m- 1 , n) + %*(m, n- 1), 
*(ra, 0) = *(ra- 1,0) + ra*(m - 2, 0) + \m{m - l)*(ra - 3, 1), 
and thence obtains 
*( 0 , 0 ) 
*( 1 , 0 ), *( 0 , 1 ) 
*( 2 , 0 ), *( 1 , 1 ), *( 0 , 2 ) 
1 
1 1 
2 1 
Then, confining himself to *(m , 0), he shows that the differential equation 
of the generating function of *(m , 0 )/m ! is 
-i — — — to — JLUj — 5 
ax 1 — x 
and consequently that 
q\X+\X* 
“ = 0^' 
This readily leads to similar results for (p(m , 1) , *(m ,2) , . 
As an afterthought (p. 335) he puts the said differential equation in 
the form 
2 ( 1 -*)^ = (2 - x 2 )u , 
dx 
and thence deduces the equation of differences 
u n = nu n _ x - j^(?i - l)(n - 2)u n _ s . 
Cunningham, A. (1874, April). 
[An investigation of the number of constituents, elements, and minors 
of a determinant. Quart. Journ. of Sci. (2), iv. pp. 212-228.] 
Cunningham devotes §§ 8-10 (pp. 220-224) to axisymmetric deter- 
minants. There is an oversight, however, in his reasoning, and his results 
are correct only as far as the fifth order. 
Roberts, S. (1874, May). 
[Question 4392. Educ. Times , xxvii. pp. 45, 66 : or Math, from Educ. 
Times , xxi. pp. 81-83.] 
Roberts’ difference-equation for the number of distinct terms is 
- u n - 1 - {n-\fu n _ 2 + %(n-\)(n-2){u n _ i + (n-S)u n _ i ) = 0, 
