570 
Proceedings of the Royal Society of Edinburgh. [Sess. 
in the usual way in partial fractions, so that we get 
1 =(x-b) h+ (xr-c) l+ .... [A 0 + A y (x - a) -\r .... +A g _ l (x-a)~ 1 + g+ ]+ 
=fa{x) +/&(*) + 
where fjx) is put for brevity instead of (x — b) h+ .... [A 0 + 
Define the linities £, £ 0 , rj, rj 0 , etc. by the equations 
. . . . i 
} (17) 
(18) 
Since 4 (* 0 , . . . . are integral functions of cp they are all commutative 
with (p and each other. Also from (16), (17), (18) the following are obvious, 
£r J = r ] €= .... = 0 , i Q rj = rjg Q = rj Q g = . . . . =0 
g + r] + .... =1 
0 — £</ = V ~ • • • • 
(19) 
Also from the equation 1 — + . . . . multiplying by f and by 4) 
we get 
£ 2 = g, £4 = 4 = 44 
rj 2 = r], rjrjQ = r) 0 = W , etc 
.! ■ 
( 20 ) 
Also cp — + r] -j- . ... )(p, and i<p = ct£+ by the definition of 4- Hence 
<p = (ag + 4 ) + (br) + rj o) + . . . . 
= g(a + 4 ) + + Vo ) + 
( 21 ) 
Conversely, if the linities 4 £ 0 , v • • • • are related to (p by the equations 
(19), (20), (21), they must satisfy equations (17) and (18); and (15) and 
(16) must also be true. From this it follows that they are definite linities 
given by <f>, in spite of the arbitrariness of the integers g + ,h+, . . . . and 
the consequent arbitrariness of the functional forms f a (x), f(x), .... To 
prove the italicised statement, we first have from (19), (20), (21) 
<p M = g(a + 4 ) m + v{b + "> 7 0 ) m + .... 
where M is any positive integer ; or more generally by the combination 
of powers such as <p M into an integral function 
/W>) =£f( a + 4) + vf(b±Vo)+ .... (22) 
where f(<p) is any integral function of <p. Giving to / the particular form 
f a defined by (17) we get [since/ a (5 + #/ 0 ) = 0, etc. on account of >y 0 /l =-0] 
/«(<£) =44(a + 4)- 
Putting x == a-\- £q in the identity 1 =ffx) + ffx) -f . . . . , we get 
1 —fa( a + 4 )* 
