262 The Rev. H. Moseley on certain Elementary Formule. 
that the series may be written under the form 
! 
l+y=1+ £ y+ terms having powers of y above the first. 
Equating, therefore, the coefficients of the first power of y in the 
two members of this equation, 
! 
1, or Z!=Z=z—12?4+128—-144+... 
Substituting in equation (4) this value of Z’, and the values 
before obtained for AR A! eg 
‘ he, Fee | SPP Zig 
(is) leas tee ae eae ee 
which is the exponential theorem. 
Lastly, to expand y in a series ascending by powers of z, 
assuine 
y= Xz + Xq2?+ Xy234+ Xyztt+.... 
* (l+y)=(14+2)*=14 X,2 4+ X24 Xge2 4 X,24+.... (6) 
Now in equation (5), let it be observed that Z is a multiple of 
z, and therefore that all the terms of the second member of that 
equation after the second, involve powers of z above the first ; 
so that the equation may be written under the form 
(1+z2)"=1+22+ terms multiplied by powers of z above the 
first power. 
Equating the coefficients of the first power of z in the second 
member of this equation, and in equation (6), we have therefore 
X,=2; and substituting this value in (6), 
l+y=14ax2+ X,2?4+ X52 + Xyz4*+.... 
*. (L+y)?=1-4 2a2+4 (2X,4-27)2? + (2X54 22X,)29 
+ (2X,+2aX54+XQ")244+....3 
but 
(1+y)?={(1+2)?}*= {1+ e+e") }%, 
*. (l+y)?=1+4 x(22 +42?) + X,(22 +27)? + X,(22427)8 
+X (224 22)44 
Equating coefficients of like powers of z, 
z+4X5=2X,+ 2? | 4X,4+8X,=2X,+2Xo2 | X.+12X,+16X,=2X,42X,+X, 
2X, =27°—»y 3X,=X,(¢— z) transposing, substituting, and reducing, 
x(a—1 x(~—1)(4—2 w(e@—1) - 9 
i ) eal —— ) X=7 ae } (a? 5a +6} 
a(a—1)(~—2)(@—8) 
X= 1.2.3.4 
