238 PHILOSOPHICAL TRANSACTIONS. [aNNO J 758. 



LXXXF. A Furilier Attempt to Facilitate the Resolution of hoperimetrical 

 Problems. By Mr. Thomas Simpson, F. R. S. p. 623. 



About 3 years before, Mr. S. laid before the r. s. the investigation of a ge- 

 neral rule for the solution of isoperimetrical problems of that kind, wherein one 

 only of the 2 indeterminate quantities enters along with the fluxions, into the 

 equations expressing the conditions of the problem. Under which kind are in- 

 cluded the determination of the greatest figures under given bounds, lines of the 

 swiftest descent, solids of the least resistance, with innumerable other cases. 

 But though cases of this sort do indeed most frequently occur, and have there- 

 fore been chiefly attended to by mathematicians, others may nevertheless be 

 proposed, such as actually arise in inquiries into nature, where both the flowing 

 quantities, together with their fluxions are jointly concemed. The investigation 

 of a rule for the solution of these, is what Mr. S. attempts in this paper by means 

 of the following 



GENERAL PROPOSITION. 



Let Q, R, s, T, &c. fig. 2, pi. 8, represent any variable quantities, expressed 

 in terms of x and y, with given coefficients, and let q, r, s, t^ &c. denote as 

 many other quantities, expressed in terms of i* and y'. it is proposed to find an 

 equation for the relation of x and y, so that the fluent of q^ -f- Rr -|- s^ -f- t^, 

 &c. corresponding to a given value of xory, may be a maximum or minimum. 



Let AE, AP, and ag, denote any 3 values of the quantity x, having indefinitely 

 small equi-differences ep, fg ; and let el, fm, and gn, perpendicular to ag, be 

 the respective values of z/, corresponding to them ; and, supposing ef = fg = x^ 

 to be denoted by e, let cm and f/N, the successive values of y, be represented by 

 u and w. Also, supposing p'/)' and ^''p" to be ordinates at the middle points 

 p'p", between e, f, and f, g, let the former p'/>' be denoted a, and the latter 

 •e"p"hy p ; putting ap' = a and ap^ = b. Then, if a and «, the mean values of 

 X and y, between the ordinates el and fm, be supposed to be substituted for x 

 and y, in the given quantity a^ -}- Rr -f s* -f t^, &c. and if, instead of .f and^, 

 their equals e and u be also substituted, and the said given quantity, after such 

 substitution, be denoted by o-'q' -f- rV" -f sV -|- t'/', &c. it is tlicn evident, that 

 this quantity aVy' -|- rV -|- sV -|- t'/', &c. will express so much of the whole re- 

 quired fluent, as is comprehended between the ordinates el and fm, or as answers 

 to an increase of ef in the value of x. And thus, if b and j3 be conceived to be 

 written for x and y, e for.f, and w (or y, and the quantity resulting be denoted 

 by Qi"q" -{- R"r" -|- s"*" -|- r"t" &c. this quantity will, in like manner, express the 

 part of the required fluent corresponding to the interval fg. Whence that part 

 answering to the interval eg will consequently be equal to aq' + rV &c. -|- a"q'' 

 -f- ti.'r" &c. But it is manifest, that the whole required fluent cannot be a max- 

 imum or minimum, unless this part, supposing the bounding ordinates el, gn 



