216 Dynamics. 



from which the value of Mm, or ds or \/dx* + dy* is readily 

 found to be as follows, namely, 



Cal. 78. , adx 



d s = 



V/2 a x a: 2 



But since the arc BM is small, a; is small with respect to , and 

 x 2 may be neglected when taken in connection with Sax with- 

 out material error, which leaves 



adx 

 d s = 



V2 a x 



Substituting this value of d s in the expression for d , we shall 

 have 



V4 Va Va VgVh a; 



or, snce 



adx . , ,. 



Now as expresses the element of an arc of a 



circle of which the diameter is 2 a and the abscissa x ; so in like 



manner, - - x expresses the element of an arc of a cir- 

 V h x a; 1 



cle whose diameter is h and abscissa a?. But the line BZ being 

 /i, if upon BZ as a diameter we describe the semicircle BM'Z, 

 M'm' will be this element ; so that we shall have 



