VOL. XXIX.] PHILOSOPHICAL TRANSACTIONS. J93 



both of fluxions and of increments, which consists in the solution of this 

 problem : having given the relations of the increments, or of the fluxions of 

 several quantities, whether they be considered with their proper integrals or 

 with their proper fluents, or not ; to find the relations of the integrals or of 

 the fluents, freed from their increments or from their fluxions. The direction 

 I have given for finding the solution in finite terms is but tentative. And I 

 must confess I know of no other method that is general for all cases. For I 

 can find no certain rule to judge in general, whether any proposed equation, 

 involving increments or involving fluxions, can be resolved in finite terms. 

 For this reason, we are obliged to seek the general solution in infinite serieses; 

 which when they break off\, or when they can any way be reduced to finite 

 terms, they then contain the solutions which we always hope for. The method 

 of finding these serieses is explained in the 8th proposition, and that is by 

 means of a series that is demonstrated in the 7th proposition. And this I take 

 to be the only genuine and general solution of the inverse methods. For 

 in this solution we always have those indetermined co-efficients, which are ne- 

 cessary to adapt the equation that is found to the conditions of the problem 

 proposed. For want of this circumstance all other methods are imperfect; 

 and particularly Sir Isaac Newton's method of finding serieses by a ruler and 

 parallelogram labours under this difficulty, because it brings no new co- 

 efficients into the resulting equation, which may afterwards be determined by 

 the conditions of the problem. However because this method is very ingeni- 

 ous and very elegant, I thought it proper to explain it in the following (viz. the 

 gth) prop. The J 0th, nth, and 12th propositions conclude the first part, 

 and in them I treat of the manner of finding the integral or the fluent, 

 having given the expression of a particular increment, or of a particular 

 fluxion of it : without being involved with the integrals; or with the fluents, 

 or with any other increments, or with any other fluxions of it. This is a par- 

 ticular case of the inverse method, but for its great usefulness I thought it 

 deserved particularly to be taken notice of. This problem is treated of in 

 general in the 10th proposition. The method of solving it in finite terms is 

 only tentative ; and when that does not succeed, recourse must necessarily be 

 had to the solution by a series in the 8th proposition. In the 11 th and 12th 

 propositions I have showed how serieses may be conveniently found, in some 

 particular cases when fluxions are proposed. 



In the 2d part I have endeavoured to show the usefulness of these methods 

 in the solution of several problems; the 13th proposition is much the same 

 with Sir Isaac Newton's Methodus DifFerentialis, when the ordinates are at 



VOL. VI. C c 



