447 



or the ratio of the fourth arc to the product of the ordinates of the 

 three preceding arcs is equal to the sum of the ratios of each pre- 

 ceding arc to its ordinate. 

 We have also 



Let x=x l =x ll , and x llt =3x, 



then we shall have Sw=4S 3 + 3S^>, 



an equation which gives the relation between two arcs of the catenary, 

 the abscissa of the one being equal to three times that of the other. 



When one abscissa is double of the other, the arcs are related by 

 the equation 2 YS=S,. 



-*-5f 



s 



and sin0= , we shall have 



S 2 Y, 1 



Y^YT+T 



an equation which enables us to calculate Y, when we know Y, since 

 S 2 =Y 2 1. Thus the catenary may be constructed by points. 



Let s, y, s^y^ s lt y lt , s tlt y ttt be four arcs and corresponding ordinates 

 of a catenary, whose abscissae are connected by the equation 



*//,=*,/ + */ + *> 

 then we shall have 



, , - 

 yy t yffn yy,,' 



The Society then adjourned over the Easter holidays to Thursday, 

 April 23. 



