104 
ME. A. CAYLEY ON THE EQUATION OE DEETEEENCES 
It may be remarked, that if a is an imaginary' cube root of unity, then the roots of 
the equation (1, 1, 1, 1, 1, l%v, 1)®=:0 are —1, <y, —&>, —c/; the differences of the 
roots are 
— —\-\-cu, — ^ 2iCt> , sy^-j-iy, 2,cu^, — 
= , — Id"*'? ^ , — 1 , — 1 , 2<y'', — iy+sy*, 
and the squares of the differences are 
, (>? , — 3*; , — Zo? , — 3 , 4iy^, 1 , 1 , 4iy, — 3 , 
from which the equation of differences is found to be 
(^q-^q-l)(^2_3^+9)(^+4^+16)(^-2^+lX^4-6^+9)=0; 
or multiplying out, it is 
(1, 6, 21, 46, 108, 546, 493, -1410, -667, -540, +12965^^, 1X»=0 ; 
which is what the preceding expression of the equation of differences becomes upon 
writing therein a—h—c—d—e-=f—\. Moreover, upon passing (as will presently be 
done) to the standard form, and then writing a—h=c=.d=:e~f—l, all the coefficients 
(except the first coefficient, which is equal to unity) should become equal to zero ; these 
two tests afford a complete verification of the result. 
The following corrections have to be made in Waeing’s result, as given by himself 
and Lagrange (Waring, Phil. Trans. 1763): — 
Waring, Meditationes Algebraicse, p. 85 — 
for -j- 169 fs read +196 q^s (in coefficient w®). 
Lagrange, Equations Num5riques, p. 108 : — 
for +1200 CE read +200 CE (in d) 
for — 169 B®D read — 196B^D (ine) 
for — 25 B® read + 25 B® (in/) 
fw + 27 read - 27 (in h). 
It may be noticed, that if in the coefficients of the several powers of ^ (as they are 
written down in the columns, without regarding the power of a which multiplies the 
entire column), we attend only to the terms independent of a, we have the series 
1, _45^ +65\ -4J®, +1^«, -^Wd, + Wf &c. 
-Wc\ - ‘IWce 
+22J®(Z^ 
— 16^V^Z 
+ 35V, 
the law of the first terms of which, up to the term +15®, is ob^ious ; but the term +15®. 
which is the last term of this initial series, is also the first term of a terminal series, the 
terms of which are deduced from the coefficients in the equation of differences for the 
