32 DISSERTATION SECOND. [partu. 



cd, that it can hardly lead to any general result. Besides, it 

 does not appear that these series can always be made to 

 involve the arbitrary or indeterminate quantity, without 

 which no fluent can be considered as complete. For these 

 reasons, such approximations should never be resorted to till 

 every expedient has been used to find an accurate wlution. 

 To this rule, however, Newton's method does not conform, 

 but employs approximation in cases where the complete inte- 

 gral can be obtained. The tendency of that method, there- 

 fore, however great its merit in other respects, was to give 

 a direction to research which was not always the best, and 

 which, in many instances, made it fall entirely short of the 

 object it ought to have attained. It is true, that many flux- 

 ionary equations cannot be integrated in any other way ; but 

 by having recourse to it indiscriminately, we overlook the 

 eases in which the integral can be exactly assigned. Accor- 

 dingly, Bernoulli, by following a different process, remarked 

 entire classes of fluxionary or differential equations, that ad- 

 mitted of accurate integration. Thus he found, that diffe- 

 rential equations, if homogeneous,' however complicated, 

 may always have the variable quantities separated, so as to 

 come under one of the simpler forms already enumerated. 

 By the introduction, also, of exponential equations, which 

 had been considered in England as of little use, he materially 

 improved this branch of the calculus. 



To all these branches of analysis we have still another to 

 add of indefinite extent, arising out of the consideration of the 

 fluxions or differentials of the higher orders, each of these 

 orders being deduced from the preceding, just as first flux- 

 ions are from the variable quantities to which they belong. 

 To understand this, conceive the successive values of the 



1 Homogeneous equations in the differential calculus, are 

 those in which the sum of the exponents of the variable quanti- 

 ties is the same in all the term 5 '. 



