240 PHILOSOPHICAL TRANSACTIONS. [aNN0 1758. 



besides that of the maximum or minimum, such as having a certain number of 

 other fluents, at the same time, equal to given quantities, still the same method 

 of solution may be applied, and that with equal advantage, if from the particular 

 expressions exhibiting all the several conditions one general expression com- 

 posed of them all, with unknown, but determinate coefficients be made use of. 



In order to render this matter quite clear, let a, b, c, d, &c. be supposed to 

 represent any quantities expressed in terms of x, y, and their fluxions, and let it 

 be required to determine the relation of x and y, so that the fluent of a x shall be 

 a maximum, or minimum, when the cotemporary fluents of B.f, c.f, D.r, &c. are 

 all of them equal to given quantities. 



It is evident in the first place, that the fluent of kx + h si* -j- ccx •\- dux &c. 

 (Z>, c, d/, &c. being any constant quantities whatever) must be a maximum or mi- 

 nimum in the proposed circumstance : and, if the relation of x and y be deter- 

 mined (by the rule,) so as to answer this single condition, under all possible 

 values of h, c, a, &c. it will also appear evident, that such relation will likewise 

 answer and include aU the other conditions propounded. For, there being in 

 the general expression, thus derived, as many unknown quantities b, c, d, &c. to 

 be determined, as there are equations, by making the fluents of Bi', c.r, oi*, &c. 

 equal to the values given ; those quantities may be so assigned or conceived to be 

 such, as to answer all the conditions of the said equations. And then, to see 

 clearly that the fluent of the first expression, Ai, cannot be greater than arises 

 from hence (other things remaining the same) let there be supposed some other 

 different relation of x and y, by which the conditions of all the other fluents of 

 B.f, c.f, nx, &c. can be fulfilled ; and let, if possible, this new relation give a 

 greater fluent of ax than the relation above assigned. Then, because the fluents 

 b^x, ccx, dux, &c. are given, and the same in both cases, it follows according to 

 this supposition, that this new relation must give a greater fluent of A^r -j- b ^x 

 -\- cc.r -f- d^x &c. (under all possible values of^, c, d, &c.) than the former re- 

 lation gives : which is impossible ; because whatever values are assigned to b, c\ 

 d, &c. that fluent will, it is demonstrated, be the greatest possible, when the re- 

 lation of :r and 3/ is that above determined by the general rule. 



To exemplify, now, by a particular case, the method of operation above pointed 

 out, let there be proposed the fluxionary quantity -—■ ; where the relation 

 of X and y is so required, that the fluent, corresponding to given values of x 

 and y, shall be a maximum or minimum. Here, by taking the fluxion, making 

 y alone variable (according to the rule) and dividing by j/, we shall have 

 ^^-4^_^ = V. And, by taking the fluxion a second time, making y alone va- 

 riable, and dividing by y, will be had — "v^iip- = v. Now from tliese equa- 



