48 PROFESSOR KELLAND ON A PROCESS 
in which form the solution has been given by Professor Booz, when 7 is a whole 
number. 
Ex. 3. Solution of the Equation for Laplace’s Functions. 
This equation has been reduced by Professor Booxs, in the Cambridge Ma- 
thematical Journal, New Series, vol. i., p. 15, to the form 
(=p) Fs Z 2(a—1) ant ln VD) Seep ee, « (0) 
where w, the quantity sought, is connected with v by the equation 
a aa (2). 
Putting w= e’ and writing D for —5 q > we have 
{(e~24—1) D (D—1) +2 (a—1) D+” (n+1)—a (a—1)}o= 
or e-2“D (D—1) vp—(D—a+n) (D—a—n—1) v=0 
or D (D—-1) e—(D—a—n—2) (D—a4+n—1) e??v=0 
Let, therefore, »=/ (-3) w (3), and this equation becomes 
D @-s(-3) w —(D—a—n—2) D-a4n=1)f (-f+1)e""m=0 
or DROS =O ete aa ee (4), provided 
f(- Ban = poet (-3) 

2 
Dia ie 1 
er oy ( D 
= Dias -2) 
ae 2 
an equation of which the solution is evidently 
| Dia tel 
D = om a5 
(DEERE 
[Sete - oto 
and by (3) v=s(-F 1% 
_D ao nerd 
z 2 2 1k ae 
Dra sit i 
ater see 

lee +n 
#s —(n—a—1)é6 | : __ g(n—a-1) 4), 
