equations of all orders , by continuous approximation. 329 
figures are found, the m th derivee will receive no augmenta- 
tion ; or, in other words, it will be exterminated when 
th of those places are determined. 
Again, when all the derivees inferior to M, the m th , have 
vanished, the process reduces itself to that of the m th order 
simply. For, the amount of the preparatory addends to L, 
the m — 1 th augmented derivee, will be m— 1 times the pre- 
vious closing addend o M ; and the preparatory addends to 
K, the m — 2 th , will be formed from its previous closing addend 
o L, by adding 0 Mr to it m — 2 times successively; a proce- 
dure obviously similar to that with which the general synopsis 
commences. 
28. From these principles we form the following conclu- 
sions, demonstrative of the facilities introduced by this im- 
provement on the original process : 
1. Whatever be the dimensions ( n ) of the proposed equa- 
tion, whose root is to be determined to a certain number of 
places, only -jj- th part of that number (reckoning from the 
point at which the highest place of the closing addend begins 
to advance to the right of that of the first derivee) needs to 
be found by means of the process peculiar to the complete 
order of the equation; after which, ~ZIT may be found by 
the process of the n — 1 th order, — by that of the n — 2 tk 
order, &c. 
2. Several of these inferior processes will often be passed 
over per saltum; and when this advantage ceases, or does not 
occur, the higher the order of the process, the fewer will be 
the places determinable by it. And in every case, the latter 
MDCCCXIX. U U 
