152 Mr. A. Ferguson on the Forces acting on a 



Several deductions of special interest may be made 

 from (vi.) : 



(a) If we put ^ = 0, we recover the equation * used in 

 the previous paper for the determination of surface-tensions, 

 viz : — 



M 1 -M = 4 W pa^K-^— y . . . (vii.) 



(/3) If we put M! = 0, we obtain the equation of equilibrium 

 of a segment of a sphere floating on the liquid, viz : — 



If in (viii.) we suppose the surface-tension of the liquid 

 to be zero, the equation of equilibrium becomes 



A? 



*/>(&*,»- 3l)=m, 



representing the state of affairs contemplated by Archimedes 

 and the elementary treatises on hydrostatics. 



(7) An interesting question is — what is the value of d x for 

 which Mx = M? Equating the right-hand side of (vi.) to 

 zero we have 



E*» = 4afR ^IL-^-n + ^V 



1 L 3 \/2a + d 1 2 3j 3 ' 



or approximately d ± 2 = 4<x 2 ; whence, more exactly, 



d 1 = 2a{l- ^ J (ix.) 



(8) Differentiating (vi.) with respect to d u we have 



^-(M 1 -M)= 2 f^f -W-2-Rd^d^ 

 ddi J 3(2a + di)f 



= (for a maximum or minimum). 



Thus the value of d x for which the pull is a maximum is 

 given by 



dl=a2 (675Ts) (x -> 



* L.c. p. 93]. 



