646] 
Or THE SECOND ORDER. 
49 
viz. we may for example write 
u 5 = l 5 4321. a 5 - 4321 (•) a s + 4321 (:) a; 
where 
4321 denotes ¿ 4 Z 3 £ 2 4 : 
in 4321 (•), we omit successively each number, viz. we thus obtain 
432 + 431 + 421 + 321 , 
= lu lÿ l 2 -j- I4I3I1 -J~ I4I0I1 I312 h • 
in 4321 (:), we omit successively each two 
numbers being 1,3; 1, 4; 2, 4, we obtain 
non-consecutive numbers, viz. the omitted 
42 + 32 + 31, 
= I4L + l s l 2 4- l s l x : 
and so on, the omissions being each three numbers, each four numbers, &c., no two of 
them being consecutive; thus in 654321 (.*.), the omissions are 5, 3, 1, and 6, 4, 2; or 
the symbol is 
642 + 531 , 
As an application, a solution of the differential equation ^ (x+ (x — a)y = 0 
is y = u 0 + u x x + u 2 x" + &c., where ri l u n = au n _ x — u n - 2 , and in particular l 2 iq = au 0 ; the 
equation of differences is thus of the form in question, and retaining l n in place of 
its value, =w 2 , the solution is u 0 = 1, u x = l x a, u 2 = l 2 l x a? — l 2 , &c. ut supra. The 
differential equation was considered by the Rev. H. J. Sharpe, who mentioned it to 
Prof. Stokes.