314 OF THE POSITIONS OF EQUILIBRIUM. 



This is the very same expression for the value of d as that which we 

 obtained in equation (226), as it manifestly ought to be, since the 

 same letters refer to the same parts of the figure ; but we have thought 

 proper to repeat the investigation, in preference to directing the 

 reader's attention to the former result ; for by this means, our per- 

 formance is rendered more systematic, and the several steps of the 

 operation are more readily traced and applied. 



Now, in the plane triangle DFC, there are given the two sides CD 

 and CF, with the contained angle DCF ; to find the side FD. 



Therefore, by the principles of Plane Trigonometry, it is 



a? 4- d 2 2dx cos.0 zz F o 2 ; 



and in the triangle EFC, there are given the two sides CE and CF, with 

 the contained angle ECF ; to find the side FE. 



Consequently, as above, we have 



2/ 2 -f d 2 2dy cos.f zz F E 2 ; 



but we have demonstrated, that according to the principles of floata- 

 tion, the lines FD and FE are equal to one another; therefore, their 

 squares must also be equal ; hence, by comparison, we have 



x z %dx cos ,(j> zz 2/ 2 Zdy cos.0' ; 

 or by substituting the value of d, equation (237), we get 



x* cos.0 V 2(6 3 +c 2 ) a 3 X zzrz/ 9 cos.f V 2(^4-c 2 ) a 2 X y . (238). 

 If both sides of equation (236) be divided by the expression #/, we 

 shall obtain 



bc(s' s) 



y =^- 



consequently, by involution, we have 



y^ j> s ~ s - 



Let these values of y and y* be substituted instead of them in equa- 

 tion (238), and we shall have 



^bc(s' s)cos.<j>' 

 xs 



and multiplying all the terms by x , we get 



6V(s' s) 2 bc(s 

 -- -- 



and finally, by transposition, we have 



