314 Transactions of the South African Philosophical Society. 



2. The result depends on Laplace's expansion-theorem in deter- 

 minants and on Cauchy's theorem that the quotient obtained on 

 dividing \a\ a\ ... a* n \ by \a° a\ ... a n ~ l \ is a symmetric function of 

 a It a 2 , ..., a n . Thus 



1 a a* « 5 a 6 



1 b M &s b 6 



1 C C* C5 C 6 



1 d fc & d 6 



1 e ei £5 e 6 



— a46404.^(<2,6,e)£*(c,cZ) -f , 



and dividing both sides by '( k (a,b,c,d,e) we have 



a^b^c^ 



(a -d)(a-e) . (b - d) (b - e) . (c - d) (c - e) 



#4&4^4 



(a - c) (a - e) . (b - c) (b - e) . (d - c) (d - e) 

 + 



The general form of the initiating alternant is 



I n /! 1 /-! 2 /^m—iyyS r/S + i y-jS + n—m—i I 



| u l u 2 u 3 ... ci m a m+1 tv m+2 ... a n \. 



When s is less than m we obtain such results as 



a 2 b 2 c 2 

 1 = ii 



= S 



(a — d) (a — e) . (b — cZ) (6 — e) . (c — d) (c — e) 



abc 



(a — d)(a — e). (b — d) (b — e).(c — d) (c — e)' 



3. Taking now an even-ordered alternant of the form 



| a°b 2 c r d r+I e s f s+I ... | 



and expressing it in terms of two-line minors, we have in the case 

 of the sixth order 



r+i n s n s-\-i 



a" a" 

 b s 



1 a a r a 



1 b b r b 1 - 



1 c c r c r+I c s c s+l 



1 d d r d r+I d s d s+I 



1 e e r e r+l e s e s+l 



1 / /'" f +l f f +I 



a*b s .?(a,b).\ c°d'e r f r+l \ - a s c s . ?(a, c).\ b°d'e r f r+I | + 



= a'b 8 . p(d, b) . {c r d\?(c,d) . ?(ef) - cV. ?(c,e) . p(dj) + ... } 



-a°c'.?(a,,c). {b*d\${b,d)..?{e,f) - b r e r . ?{b,e) . ?(d,f) + } 



+ 



