

116 Mr. T. Muir on the General Equation of 



an expression for u x in terms of iii and u . Professor Cayley 

 calculates in succession the required expressions for u s ,u^u 5) u 6 , 

 and thus by induction arrives at a simple and elegant law of 

 formation, by means of which one is enabled to write down 

 the coefficients of u x and u in these expressions with great 

 ease. 



The object of the present note is to establish rigorously 

 Professor Cayley's law; to connect it with a class of functions 

 (first dealt with in the Philosophical Magazine*), and so give 

 it a wider sphere of usefulness; and to indicate in what direc- 

 tions and with what profit Professor Cayley's result may be 

 generalized. 



2. From the given equation we have 



— u & + a 5 u 5 + &4W4 = 0, " 



—u h + a±u± + b z u % =0, 



— w 4 + a d u B + b 2 u 2 = 0, y 



—u s + a 2 u 2 + &i% == 0, 



- u 2 + aiUi + b u Q = ; 





and hence 







u 6 = 



a 5 b 4 ... 



— 1 a 4 b z . 

 . — 1 a 3 b 2 . 

 . — 1 a 2 b^i 



. —1 aiUi + b u 





= 



a 5 b 4 . . . 



U\ + 



a 6 b± 





— 1 a 4 b 3 . . 





— 1 a 4 & 3 . 





. — 1 a s b<z . 





. — 1 a 3 b 2 . 





. — 1 a 2 \ 





. — 1 a 2 





. — 1 a 1 





. —1 b u 



= K ( h b 3 h b x \ +bK ( ^ b z b 2 \ 

 \a 5 a 4 a z a 2 a l J x \a 5 a 4 a 3 a 2 ) v 



If now we use Professor Cayley's own theorem regarding 

 the expansion of a determinant according to products of the 

 elements of the principal diagonal, we obtain at once the 

 following as the rule for calculating the coefficient of u v 



Write down as the first term 



a h a 4 a z a 2 a x : 



in every possible way leave out of this one consecutive pair of a's 

 and insert instead the b that in the continuant notation stands 



* 1853, vol. v. p. 453. 



