128 Mr. Boor on the Analysis of Discontinuous Functions. 
As regards the expression of discontinuity, this formula is equivalent to (6); but 
we have gained this important point, that 2 only enters the first member in a 
rational form. 
The general formula (9) may be verified thus, for the particular case of 
W/() son 
ka'da 2hra— Se a 
ka" C da. 
? (a—ay a) sa = ka te a ee eae 
ka"da 
= u,, then the above equation gives, on integration, 
Let |" aaa 
1 > 
ul, == ) ka" da + 27u,_.—2°Uy,_,—k’U,_,s 
™ Jp 
which, when & becomes infinitesimal, gives 
ln — 2LUn_, BUy_» = 0, 
an equation of differences, of which the solution is 
Uy, = (c+ e'n)a". 
To determine the constants, we have 
Uy Ge 
u, = (c+ ec')e. 
Hence 
U 
/ I 
C=bh, € = Fm 
therefore 
Un 
i= (w, a ae _ wn) Tie (11) 
Now 
u, = 1, or 3, or 0, 
x 
U, = 4, OF 5, or 0, 
according as 7 lies within, upon, or without the limits p and g. Substituting in 
n 
: ; a, : 
(11) we have in the first case wv, = 2”, im the second wu, = yin the third v,= 0 
as we should expect. ' 
