252 
[755 
755. 
ON THE MATRIX 
( a, b ), AND IN CONNEXION THEREWITH 
( c, d I 
ax + b 
THE FUNCTION 
cx + d ’ 
[From the Messenger of Mathematics, vol. ix. (1880), pp. 104—109.] 
In the preceding paper, [due to Prof. W. W. Johnson,] the theory of the symbolic 
powers and roots of the function ^ + ^ developed in a complete and satisfactory 
manner; the results in the main agreeing with those obtained in the original memoir, 
Babbage, “ On Trigonometrical Series,” Memoirs of the Analytical Society (1813), Note I. 
pp. 47—50, and which are to some extent reproduced in my “ Memoir on the Theory 
of Matrices,” Phil. Trans., t. cxlviii. (1858), pp. 17—37, [152]. I had recently 
occasion to reconsider the question, and have obtained for the nth function <j) n x, where 
(f)X = aX + ^ , a form which, although substantially identical with Babbage’s, is a more 
COC “p CL 
compact and convenient one; viz. taking \ to be determined by the quadric equation 
(A. + 1) 2 (a + d)~ 
ad —be’ 
the form is 
( (V*-* - 1) {ax + b) + (\ n - \) (- dx + b) 
^ (\ n+1 — 1)(cx + d) +(A. ,i — X.)( cx — a)' 
The 
matrix ( 
question is, in effect, that of the determination 
a, b ); viz. in the notation of matrices 
c, d | 
(«i, 2/i) = ( a, 6 ) {sc, y), 
\o,d\ 
of the ?ith power of the