82 STATICS OF SYSTEMS OF PARTICLES 



the point o. Since s denotes the arc of the curve measured from 

 0, we have s = at the point 0, and therefore the y coordinate 

 of (putting s = in equation (22)) is 



y = c + a constant. 



Let us agree that Oo is to be made equal to c, so that y c at 0. 

 Then the unknown constant of integration must be zero. Thus 

 equation (22) will be y^^ + c 2 . ^>3) 



The last step is to transform the variables from y and s to y and x. 

 The relation which enables us to do this is obtained by eliminating 

 6 from relations (21), and is 



(ds) 2 = (dy? + (dx)\ (24) 



The equation already obtained is 



s = V?/ 2 c' 2 ' 



so that ds = / y y 



^/f-c* 



and on eliminating ds from this and relation (24), we obtain 



From this, (dxf = (dy? \~^ z - 1~| 



f-<? 



Integrating this, we obtain 



so that dx= . Cdy - (25) 



/ o o \ 



-> (26) 



c c 







x - -- 



where cosh - = ^ (e c -f e c ). 



c 



The student who is not familiar with the hyperbolic cosine (cosh) func- 

 tion will easily be able to verify equation (26) by differentiating it back 

 into equation (25). 



