TABER. — SCAIAR FUNCTIONS OF IIYI'ER COMPLEX NUMBERS. 653 



For any mimher of the quadrate €„„(//, /• = 1, 2, ... 7i) the two 

 scalar fuiK'tions with resi)oct to this miinhcr system defined in theorem I 

 are e([ual as shown in § 2; and, therefore, but a single symbol is re- 

 quired for these functions. I sliall denote by N .1 the two equal scalar 

 functions of an\- numl)er 



(65) 



11 n 



A = T. T. <luv€uv = 



U=l 1=1 



On, fli2, . . . flin 

 f''21» f'22) • • • ft'in 



of the quadrate; and, by theorem I, we then have 



(66) Siuu = - , Stuv = {u, v = 1, 2, ... n; v n^ xi)}^ 

 n 



and, therefore, 

 (67) 



S.-l = X 21 «ui;'Sei,„ = Y^ a 



u=l v=X 



n 



u=\ 



I shall denote simply by 1 the modulus of the quadrate, and pi, for 

 any scalar p, simply by p. We have 



n 



(68) 1 = Z e„,. 



u=\ 



n n 



Any number -1=2] Y. ^u^^uv of the quadrate satisfies an equation 



U=l 1=1 



(69) <t> U) ^ (A - pi) {A - p,) . . . (.i - pn) = 0, 

 where the p's are scalars; and we have 



(70) (/>(p) 



P — aib — ai2, • • • 



— O-i], p — Ct22> • • • — Chn 



Chn ! = (p — Pi) (p — p-i) 



... (P-P.).2° 



— flnlj — f'n2> . . . P — (Inn 



19 For the number of linearly independent numbers X of the quadrate 

 satisfying the equation CuuX = X is /(, since every such number is linearly 

 expressible in Cuu ^ui, ■ ■ ■ fun, and each of these numbers satisfies this equa- 

 tion. Therefore, by theorem I^ n-Si€un = «. Similarly, the number of 

 linearly independent numbers X of the quadrate satisfying the equation 

 Xeuu = A' is a!.so n; and, therefore, iC-Soeuu = n. Since, for v 7^ u, e^ is nil- 

 potent, ^'itur = '"^iiuv = (v ^ u). 



20 Cayley: Pliilosophical Transactions, p. SOO (1858). 



