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termined from a comparifon of homologous terms is, perhaps, the 

 befl where it can be pradiled ; yet the cafes are very numerous 

 where without other afliflance it is difficult and almoft impoffible 

 to pradice it with any advantage. A method, therefore, which be- 

 fides being in all cafes as fimple as any of the others is as general 

 as can be defired, and is often attended with the fuperior advan- 

 tage of demonftrating the law of the feries, muft be an objedb 

 for the confideration of mathematicians. Such a method is at- 

 tempted in the following pages. Its foundation is built upon a 

 theorem firft given by that excellent mathematician Dr. Brooke 

 Taylor. This theorem, given in Cor. 2. Prop. 7. page 23, of his 

 method of Increments, is well known, and is in purport as 

 follows : 



i 

 If X. and z be two variable quantities, the relation of which is j 



given, then while x by flowing uniformly is increiafed by x, z .1 



.. \ . 



z z 



will be increafed by ^ + 1 H &c. In which the va- 



^ I. 2. I. 2. 3. 



lues of 2r, z, &c. are to be determined from the given equation. 



It readily occurs that this theorem contains a method of de- 

 riving the values of one quantity by a feries afcending by powers 

 of the other: and accordingly feme authors have ufed it in a few 

 limple cafes, but have not attempted a general ufe. And upon 

 confideration it is obvious that without farther affiftance it cannot 



be 



