equations of all orders , by continuous approximation. 313 
here employed are lower by one order than those which 
naturally occur in transforming an equation of the n th degree. 
But it is much to be wished, that these coefficients could be 
entirely dispensed with. Were this object effected, no mul- 
tipliers would remain, except the successive corrections of the 
root, and the operations would thus arrange themselves, in 
point of simplicity, in the same class as those of division and 
the square root. 
12. Nor will this end appear unattainable, if we recur to 
the known properties of figurate numbers ; which present to 
our view, as equivalent to the n th term of the m th series : 
1 . The difference of the n ,b and n — 1 th term of the m - f- 1 th 
series. 
2 . The sum of the first n terms of the m — i tk series. 
3. The sum of the n th term of the m - — 1 th , and the n — 1 th term 
of the m th series. 
The depression already attained has resulted from the first 
of these properties, and a slight effort of reflection will con- 
vince us that the second may immediately be called to our aid. 
13. For this purpose, let the results of Art. 9 be expressed 
by the following notation : 
+ AV 
A , = A j +B ' r' 
B'=B + CV 
C — C 3 + DV 
V' = V + U 'r' 
U'= U +r' 
the exponents subjoined to any letter indicating the degree 
S s 
MDCCCXIX. 
