342 Account of certain Improvements 
(4), a5 } n(a + b+cy—d4 (a+ b+ c)"—2d? 4. ae 
(a+b+c)"—*d3 + &e, } 
n.n—|} 
"Nath tetd) 24 
(a+b+c+d)e+ &e. } 
(5), + §n(a+b4+c+4+d)"—e4 
t 
n.n—\Ln—2 
1.2.3 
(6), + &e. 
Demonstration. 
Calling each series inclosed within each two braces a term, 
a” being the first term :—I observe that the first and second 
terms are equal to the expansion of (a+L)” ; that the third term 
is the expansion of [(a+) +c]” considered as a binomial want- 
ing the first term (a+ 0)” ; that the fourth term is the expansion 
of [(a4+b+c)+d]” wanting the first term (a+b+c)”, and so 
on: therefore the whole expanded series is equal to 
(a+b)” + : (a+b+c)” —(a+1)n} oe ) (a+b+c+d)" — 
(a+b+o)n} = ; (a+b+e+d+e)r—(a+b+ce+4d)"} +&e.= 
(a+b4+ct+td+te+&c.)” 
And thus we have another general rule for raising any number 
to the mth power, besides that of multiplying the number con- 
tinually by itself (7—1) times. 
I shall here present the reader with a numerical example or 
two in involution, in order to explain the nature of evolution : 
for this purpose we have 
(37658)” = (30000 +7000 +600+50-+4 8) 
therefore (37658)" =(30000)” &c. 
- } 2(30000)"— (7000) +" (30000)"—2 (7000)? + &e. i 
1.2 
++ } 2(37000)"—(600) + ““— (87000)"*(600)* + &e. } 
+ {n(87600)"— (50) +“ (37600)"-*(50)*+ &e. } 
++ } n(37650)"— (8) + “= (37650)"-*(8)? + &e.? 
Or universally thus: 
Since any scale of numbers may be generally represented by | 
ax™ + bam—l 4 cym—24 ,.,4+hkx+1 we shall have 
(aam + bam) 4. cym—2 Loi ae ae +tko+tl)n = (axm nr 
+ } m(axm >t (ba) rae (ax)? (ba™—1)2 4. &e. ' 
+ 
