113 
1920-21.] A Continuant of Cayley’s of the Year 1874. 
we find that 
fti ~ *1 - 0 
ft 2 _ ftl^l + 27T 2 = 0 
ft 3 ~ ft 2^1 + ftl^ ~ 3?r 3 = 0 
fti ~ fts 77 ! + ft 2^ 2 ~ ftl^S + 4tt 4 = 0 
and on solving for the 7r’s there is readily obtained 
whence 
02 ftl 2 • 
A P 2 ftl 3 
A ft 3 A A 
{[>> + [%}*•= 
ax + fyy 1 
ax 2 + by 2 ax + by 2 
ax 3 + by 3 ax 2 + by 2 ax + by 3 
ax 4 + by 4 ax 3 -f 6?/ 3 ax 2 + &?/ 2 ax + by 
T 
— the new result referred to. 
(6) The most interesting point about this is that the relation on which 
it depends, 
ftr — ftr—i^i + ftr~ 2 7r 2 ~ • • • + ( - 1 ) r r7r r = 0 , 
is exactly Newton’s relation between the s’s and the cs of a number of 
quantities, that is to say, where s r is the sum of the r th powers of the 
quantities and c r the sum of the r th combinations of them. 
(7) Turning now to Cayley’s second form we shall first show how there 
may be derived from it a better, especially as regards the interchange of 
a, as with b ,y. Using the fact that any factor of an element of a con- 
tinuant may be removed and attached to the conjugate element, we obtain 
in the case of the fourth order 
(a -3)x-\-by x 
(a-3)(x — ■*/) (a— l)x + (b - \)y 2x 
. (a — 2)(x - y) (a+ l)x+ (Z> - 2)y 3x 
(a - l)(x - y) (a + 3)x + (b — 3)y , 
which for shortness’ sake we may denote by 
K 4 (a — 3 , x , b,y), 
the a — 3 under the functional symbol being written instead of a to recall 
the element in the (1 , l) th place. If in this we diminish the second row 
by the first, the third by the new second, and so on, we obtain 
VOL. XLI. 
8 
