VOL. L.] PHILOSOPHICAL TRANSACTIONS. 23Q 



to remain the same, is also a maximum or a minimum. Hence, in order to de- 

 termine tiie fluxion of this expression o^q + r'/ &c. q!'c^' -j- ^'r" &c. which must 

 of consequence be equal to nothing, let the fluxions of a' and q, taking » and u 

 as variable, be denoted by Qa and '^u\ also let Ra and rw denote the respective 

 fluxions of r' and r \ and let, in like manner, the fluxions of a'', q\ ^\ r", &c. 

 be represented by q^, q'w, k^, 1-w, &c. respectively. Then, by the common rule 

 for finding the fluxion of a rectangle, the fluxion of our whole expression aq -f- 

 R'r &c. + Q.''q'' + Vi'r" &c. will be given equal to o^'qu + 9' q« + r7m -{■ r'T<a &c. 

 + a'g^ 4- q"li& + R' ~KJ + "" K^ &c. = O. 



But u-\- w being = gn — el, and p — a = 4- (gn — el) constant quantity, 

 we therefore have ^ = — «, and ^ = «: also m being = 2 r/)' = 2 a — 2, el 

 thence will M = 2 a: which values being substituted above our equation, after 

 the whole is divided by «, will become 



2Q'^ + <7'Q + 2RV + r'H, &c.~2Q"f+ /q— 2r7+ r'R&c. =0; 

 or, a'7 — a'^ + r'^ T — r7 &c. = J- (<^'q + ^' q) + 4. (/ r + r^' r,) &c. 

 But a"^ — a'y, the excess of (k"q above q'J, is the increment or fluxion (an- 

 swering to the increment or fluxion .r) arising by substituting b for a, (3 for a, 

 and w for u. And, with regard to the quantities on the other side of the equa- 

 tion it is plain, seeing the difference of 9' q and q"'^ is indefinitely little in com- 

 parison of their sum, that q'q may be substituted instead of 4. (^' q -j- q"'^,) &c. 

 which being done, our equation will stand thus : 



Flux. Rq -f r'7 &C. = ^'q -f- t'r &C. 



But /q + /r &c. represents (by the preceding notation) the fluxion of /a'-f- 

 / r' &c. or of Qiq -f Rr &c. arising by substituting a for y, making a alone 

 variable, and casting oflf a . If therefore that fluxion be denoted by v^ we 

 shall have flux, o-'q -f- R'r &c. = i;, and consequently ofq -f- r^ &c. = v. 

 But Qi'q + R'r &c. (by the same notation) appears to be the fluxion of <a'o' -{- 

 rV &c. or of Qiq -f- Rr &c. arising by substituting u for y, making u alone va- 

 riable, and casting off" u . Whence the following 



general rule. 



Take the fluxion of the given expression (whose fluent is required to be a 

 maximum or minimum) making 3^ alone variable; and, having divided by y, let 

 the quotient be denoted by v. then take again the fluxion of the same expresssion 

 making y alone variable, which divide by j/; and then this last quotient will 

 be = y 



When y is not found in the quantity given, v will then be = O ; and conse- 

 quently, the expression for i, equal to nothing also. But if z/ be absent, then 

 will t; = O, and consequently the value of i; = a constant quantity. It is also 

 easy to comprehend that, instead of y and y, i and x may be made successively 

 variable. Also, should the case to be resolved be confined to other restrictions 



