296 
Transactions of the Royal Society of Sottth Africa. 
where U, Y, W stand for 
o o o o 
respectively.* In the next place, since it is readily verifiable that 
(St^i + ^h) (2^1 - = U + 2V + 2W, 
we may strike out from the three determinants of the equality the factors 
+ ^a^ — U + 2Y + 2W respectively, thus arriving at 
M N ^ C (U, Y, W, W, Y) 
+ ■ - U + 2Y + 2W 
and therefore by using the result specified in §1, and by factorising 
C (U, Y, W, W, Y) with the help of e, an imaginary fifth root of 1, we 
obtain 
V J ' = (U + Ye + + Wa3 + Ya4) 
whence 
M 
+ 
(U + Yt- + We^ + Wf + Y£3) 
(U + Ya3 + We + Ws* + Y^s) 
(U + Ya* + Wb^ + We3 + Ya ) 
{U + Y (a + a^) + W (a3 + ,^)}^ 
{U + Y(a2 + E3) 4- W(a + a^) }2 
{U + Y(a + a-i) + W(a2 + ^-2)} 
{U + Y(a2 + ^-2) + W(a + E-1)}, 
and so, finally, the interesting result regarding the resolution of M into 
factors, namely 
+ ^2+^2 % + ^3 «4+^4 ^5 + ^5 
a^4 4-?>3 ^5 + 64 «iH-?>5 %-r-&i ^^3 + ^2 
+ «^4 + ?>5 ft5 + ?>i + ^2 + ^3 
^2 + ^5 ^8 + ^1 «4 + ^2 ^-^-^S ^^1 + ^4 
(2a.i2_S6i2) + 2{ia^a^-\\) cos ^ + 2(ia^a^-±h^b^) cos ^| 
. -USV-^V) + 2(iaia2-i&^) cosi^ + 2(±a^a^-ih^h) cos ^""A 
i 5 5 J 
the generalisation of which for any order is enunciated in the ' Messenger of 
Math.,' xi, pp. 105-108. 
O - o 
* S is used as the symbol of a cyclic sum, for example, Sajaa = a^a^ + a^a^ + a^a^ 
+ a^ai + ar^a^. 
RONDEBOSCH, S.A. ; 
February 22, 1920. 
