326 



ISOPERIMETRICAL PROBLEMS. 



Jwptrime- 



trical 

 Problem!. 



tion, of precisely the same form, 



Q.,+Q,.=0, orQ.+(Q+dQ>=0. 

 Eliminating, then, one of the variations, and dividing 

 bytheother,^ ^ of dP_dQ =Q 



the quantity V, under theintegral sign, involves one or 

 more indeterminate integrals, or expressions given by 

 an unintegrable differential equation, as well as where 

 the function to be made a maximum is not merely a 

 single integral, but a function of one or more com- 



fxds 



bined ; as, for example, where is to be a maxi- 



Jdi 



mum,J*ds being given; but of these the necessary limits 

 of this article preclude any thing beyond this historical 

 notice. It now only remained to discover some general 

 method of treating isoperimetrical projalems, which 

 should dispense with the resolution of integrals into 



whose complete integral is 



P + oQ = (a) 



a being arbitrary and constant. Now, P. > is the 



variation of /".V d x, on the supposition of one ordi- 



nate.y only varying, and Q. , that offWdx; hence, 



(P+aQ) is that of fVdx + a. fWdx, so that the their elementary portions, and reduce their "treatment 



equation () which sati'sfies the relative maximum, to a regular series of purely analytical processes. This 



is the very same to which 'the treatment of *-" "m-IM hv T.n^no-P fm tw ,. 



fV d x^-a. fY? dx as an absolute maximum would 



have led. 



A reasoning precisely similar applies, whatever be 

 the number of properties concerned. Suppose them 

 rVdx,fWdx > fZdx,&c. and ,, a, *, &c. being the 

 variations of the same number of consecutive ordinates, 

 each of the conditions indiscriminately gives an equa- 

 tion of the same form. 



P. 1 + P,. a + P!. T + &C. = 



Q., + Q,. + Q,.*- + &c. = 



And it is evident that the equation 



P + a Q + b R + &c. =? 0, 



where a, b, c, &c. are arbitrary constants, will be the 

 complete integral of the differential equation resulting 

 from these by the elimination of , a, *, Sic. because it 

 gives 



P, + a Q.+6 R, + &c.=0, &c. 



and multiplying these equations in succession by 

 , a, v, &c. and adding them together, their sum will 

 vanish, as is readily perceived, by reason of the former 

 system of equations. 



The problem of relative maxima and minima is thus 

 reduced at once, and in all its generality to that of ab- 

 solute ; for the preceding demonstration applies to all 

 cases. We need only add to any one of the properties, 

 the others, multiplied each by an arbitrary constant, 

 and then make the sum an absolute maximum or mini- 

 mum. As a single iastance, let us inquire what curve, 

 under a given circumference, contains the greatest area, 

 or in vrhichj~d x V \ + p* being given, J"y d x shall 

 be a maximum. Here the quantity to be made an ab- 

 solute maximum isJ'(y + a</\ + f}dx; and V in this 

 case being a function of y and p only, the iquation (B) 

 holds good, and gives 



Isoperimc- 



tllfitl 



Problems. 



desideratum was supplied by Lagrange (in two me- 

 moirs, published in the Melanges de Turin, tomes ii. and 

 iv. 1760, 1770) by one of his first and greatest disco- 

 veries, the calculus of variations, and with such com- 

 plete success, that nothing beyond it can be expected in 

 future. Euler, with a memorable candour, of which, 

 perhaps, no parallel instance can be produced in the an- 

 nals of science, and which forms a striking contrast 

 with the passages we have had occasion to notice in the 

 early history of the subject, was the first to acknow- 

 ledge its superiority over his own methods ; and, dis- 

 daining those feelings of jealous rivalry the circum- 

 stances of the case were so peculiarly calculated to ex- 

 cite, hastened to become the commentator of the new 

 method, and to substitute it in the place of his own. The 

 principle of the calculus of variations, and its connec- 

 tion with the present subject, has been purposely kept 

 in view in the preceding pages, and will have been 

 already been made sufficiently evident. For the detail of 

 its processes, and an account of its successive improve- 

 ments, as well as for the demonstration of the theorem 

 we shall require, the reader is referred to the article 

 Calculus of VARIATIONS. 



The essential distinction between this mode of treat- 

 ing isoperimetrical problems and Euler's is this, that 

 the latter estimates the change produced in an integral 

 fV dx by the variation of -one, or, at most, a limited 

 number of ordinates ; whereas the former considers the 

 change effected in it by the continuous variation of the 

 ordinate along the whole extent of the curve, infinitely 

 small indeed in quantity, but regulated by a perfectly 

 arbitrary law. The variation ofjfV d ,r is determined, 

 which it undergoes, not by the change of one element 

 only of the curve, but by the whole curve ABCD (Fig. 5.) 



Fig. &. 



which reduced, affords 



and integrating, 



y C 



the general equation of a circle. 



In the work above cited, Euler extended his re- 

 searches successfully and correctly to the case where 



undergoing an infinitely slight change in its nature, and 

 passing into another abed, infinitely near it, but 

 whose deviation from it in the different parts of its 

 course is in all other respects perfectly indeterminate 

 and arbitrary. It is demonstrated in the article referred 

 to, that (supposing 3 x = 0) the whole variation so pro- 

 duced ( J/*V d x) is reducible to the form, 



where A,, and A , are the values assumed by a function 

 A,, (in which 2y, d 9j/, &c. are not included under any 



