in Involution and Evolution. 343 
+ {nf (ax™) +(a"=!) |" (co) +P (an) + 
(La) | ear) 4+ ee i +&c. for the expansion 
of any power of a number, according to any scale of notation. 
But perhaps it will be more useful to descend to some parti- 
cular instances by which the method and principle may be seen 
to advantage. Therefore let a+)+c+d+e+&c. be raised to the 
cube; then will (a+l+c+d+ei=a+ 1 30h +3ab? +L) ' 
-F {3(a+b)%c+3(atb)e+o} “ye }3(a+b4c)d+ 3a4)+0) 
2 2 
+d § 
+ {3(atb+ctd)et+3(ath+c+dje+e. 
Or, because that in any term the members constituting that 
term have a common factor, the same may be exhibited thus: 
(at+b+c+d+ei=ai+h } 8a7+3ab+L? | +¢ )3(a + b)?+ 
3(a+b)c+cr} 
+d {3(atb+c)*43(atb+o)d +d} +e(B(atb+e+d)?+ 
3(a+l+c+d) e+e } 
Let it now he required to find the cube of the number 5436. 
Here a=5000, a+l=5400, a4-b4+c=5430, anda+b+c+d= 
5436; consequently =400, e=30, and d=6. 
Operation by the method here demonstrated. 
a@ = 125000000000 
3a2b= 30000000000 
lat Sal 2400000000 
nel ce 64000000 
3(a+b)*c= 2624400000 
2a) Salem iassoono 
<2 Hse = 27000 : 
3(a+b+c)7d= 530728200 
Bd.) Sache vsncs0 
Piatt Pz 216 
160634321856 
The number of figures in this operation may be considerably 
lessened by considering that the first period would be the same as 
if it consisted only of 54=50+44; and the second period as if it 
consisted of 543=540.+3; and the third period as if it consisted 
of 5436=543046. Therefore the operation in this form is 
Y 4 a= 
