[ ^^3 ] 



be pradJifed in cafes at all complex. For if in an equation exprefllng 

 the relation of x and « the fucceffive fluxions be derived one from 

 the other generally and without having regard to particular values 

 of X and z, almoft infuperablc trouble vpould arife except in the 

 moft fimple cafes, and the method be very far inferior to others. 

 This farther afliftance I have endeavoured to give in the following 

 pages, principally by theorems for taking fluxions of different orders 

 per faltum, that is, without finding the fluxions of the inferior 

 orders. Thefe will render the theorem of Taylor of the moft ex- 

 tenfive utility, as will beft be feen by the examples hereafter 

 given. 



M. De la Grange is the only author I know of who has at- 

 tempted to Amplify the computation of z, z, &c. This he has 

 done by a moft elegant theorem for an equation of a particular 

 form (See Coufin's " Aftfon. Phyfique, Art. 20, p. 15.") But no nfe 

 can be made of this theorem except in equations of that particular 

 form. The theorems for taking fluxions per faltum will enable 

 us to compute the values of z, z. Sec. by fubftituting the va'ues of 

 X and z when they begin to flow, and as in that cafe it ofien 

 happf=ns that the problem is fuch that x and z begin from nothing, 

 the conclufions are then derived in the moft Ample manner. 

 * 



The method of aflTuming a feries with undetermined coeflicients 

 for ihe quantity to be found, befides the objeclioas in every par- 



S s 2 ticular 



