186 Transactions of the South African Philosophical Society. 

 The equations are best written in the form 



-ft-s + 2* = 4/3 2 -3/3 3 \ 



ft + s = 4ft + 3ft - 4ft 



-ft -2s + 4* = 4ft +4/3 4 -5/3 5 



ft + 2s =6/3. + 5ft -6ft )■ 



-ft -3s + 6* = 6/3. + 6ft -7ft 



ft + 3s = 8ft + 7ft - 8/3 8 



• / 



and a little examination of them shows the need for considering 

 two cases, viz., n even, and n odd. 



4. Let us consider first the case where n = 2m. Here the last 

 /3 is ftm-i, and the last equation 



ft + (m - l)s = 2mft + (2m - l)ft w _i. 



Taking the first equation of the set, an equation derived by addition 

 from the first two, an equation similarly derived from the first three, 

 and so on, we obtain the following equivalent set 



-ft-s + 2t = 4ft -3ft, 



2t= 8/3 2 -4ft, 

 - ft - 2s + 6>t = 12ft - 5ft, 



6£ = 18ft - 6ft, 

 -ft -3s + 12* = 24ft -7ft, 



12* = 32/3 2 - 8ft, 



2(1 + 2 + ... + m-l)t = 2m 2 ft 

 From the last of the set we have 



m-1 



2m 



and substitution of this in the others gives 



0s = J(/j.+« -— <). 



O \ TO 



TO- 2 



ft = J (ft + 2s - ?j?*Y 

 & = i (ft + 3s - y A 



r4 



2m 



ft = 



m - 3 , 

 2m 



ft = 



m - 4 / 

 2m ' 



ftm-i = ~ 1 (ft + W- l.S -m-1.*), ftm-2 = a-*. 



2m - 1 2m 



