Definite Integrals, with Physical Applications. 383 



Make the middle point of the hypothenuse, (the length of 

 which we may suppose = 2,) the origin of co-ordinates; the axis 

 of x being the hypothenuse itself. 



The Equation of ACa, generally, is y = l+x 



...of BCb is y= 1—x, 



and therefore the general equation of the system is 



(y-x-l) (y + x-l) = 0. 



in which, any value of x, as OP, corresponds to two values 

 of y as PQ, Pq; the least of these two values is that which 

 belongs to the black part ACS, at both sides of the origin ; 



the equation peculiar to this part is therefore y = the least of 



{1 — x 

 substitute in Art. (16). 1—x for a, and 1 + x for 8, and we 

 1 +x 



get for the required equation 



y=i (I-* 1 ) + jo ■ fr-*? + or! • (1 "*' 2)3 + &c ' 



We may now treat this value of y, by the ordinary methods 

 of analysis, for the attainment of any object which has sole 

 reference to the division ACB of the given system. 



Thus to find the area of the triangle ACB, we must take 

 as usual f x (y) from x= — 1 to #=+1. 



Now the general term in the value of y is 



1.1.3 .5 (2»- 



2.4.6 .8 2m 



the integral of which from x= -1 to x= +1 is 



2 1 1 



or 



(«»- !).(«» + 1)2»-1 «»+!' 



