320 
Mr. Robertson's Demonstration 
1 1 _ j j 
#th power, therefore, of the series x r + yzx r -f 7 x ~ ~ 
z x 1 
+ 7 * 
z 3 x r + , See. is equal to x 
+ - zx T + - xl 
1 z*x r 
. ft — — * 7 — z . 
+ — X I X z' 
x r -f > & c - an d series, when n is equal to r, becomes 
equal to x -f z. Hence x -f z 1 r = x r -f yzx r + 7 x 
1 2 
_ Z* X ’ 
+ T* 
3 
sequently x-\-z v = x r -j- yzx r -f 7 x - 
+, &c. and con- 
+ 
z x 
— ~ 3 
' 3 - +, &C. 
z X' 
21. From the preceding method of investigating the theo- 
rem it also follows that x — %l r = x r — — zx r +7X L 
z 1 x T — 7x1 x- z 3 x r 3 +> &c. For the series 
^2 2 
« 7 
7 zx* 
3 
1 , n_ v 7 — 1 r “ 2 1^7 — 1 w 7 — 2 
' r 2 r 2 3 
Z X 
-{-, &c. being multiplied by j- 
ZX‘ 
+ -x 
-f, &c. the pro- 
I z x r — — x r - , x -- . . z 3 x r 3 
2 r 2 3 
duct will stand as below, the laws of arrangement being the 
same as those established in the 14th and 15th articles. 
