546 J. D. HAMILTON DICKSON ON RELATIONS BETWEEN FUNCTIONS 
P, becomes (w—a)(w— B)+(w—a)(v—y)+ ...15 terms 
= 1527’°-5P,2+P, 
= , ler ee & PaBeh yeleer | Re TTSS eee as 
3 . 
P, becomes (x —a)(w@—f)(w—y) + (w—a)(wx— B(w—8) + ... 20 terms 
202°—10P,2° + 4P,7—P, 
20 (ax? + 3622+ 38cex+d : ; ; ; : (20), 
a 
so that, in general, we should find, instead of each P, all the terms of the sextic 
up to and including the term corresponding to the P in question, each term 
multiplied by the Binomial coefficient of its place in the expansion of the power 
the index of which is the same as the suffix of P; and the whole multiplied by 
the Binomial coefficient of the sixth power corresponding to the P considered, 
and divided by a. 
The process is general, and, mutatis mutandis, is suitable for an equation of 
any degree. 
Now, let such an expression as az’ + 3ba’ + 3cx +d be denoted by the symbol 
(d). Thus, the sextic in question would be (g)=0. Such a symbol implies 
Binomial coefficients. The Jetter will also, from its place in the alphabet, indi- 
cate the degree of the expression (or equation). This notation harmonises with 
that employed before, when.x was zero. 
Hence, writing (y) for v-—y+2—8+a2—e+2—© (ys) for (vx—y)(w—8) + ..., 
and so on, 
1 5105 10" baal 
w {(a—B)| 1, (y) » (79), (e) » (v8eb) , } =6@),(0),(@),(&), FP)» ee 
-> 1, (y) » (78), (y8e) , (ySel)! (a), (0), (6) (2), (2) (FP) 
(0) , 5(c) , 10(d) , 10(¢) , 5(f) , ye (21). 
(a) , 5(0), 10(c) , 10(d), 5(@)_ (Ff) 
6 
For example, 
wd {(a—B)'(a—y)(#— 8) + (2—y)(@—¢) + (%@—7),(@—€) + (@—8)(@—€) + 
(w— 8)(v—£) + (w@—e)(v—8) I}, 
(2) , (2) 
(a) , (d) 
ax+b , au*+4ba’ + 6ca’?+4dxr+e|. 
a@ , x°+4+3ba°+3ca+d 
= 360 
060 
