When a body is immersed separately into three liquids of specific gravities `S_(1),S_(2)andS_(3)`, its apparent weights become `W_(1),W_(2)andW_(3)` respectivley. Show that `S_(1)(W_(2)-W_(3))+S_(2)(W_(3)-W_(1))+S_(3)(W_(1)-W_(2))=0`.
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Let the real weight of the body be W and the volume of the body be V. `therefore" "W_(1)=W-V*S_(1)" "...(1)` `" " W_(2)=W-V*S_(2)" "...(2)` `" "W_(3)=W-V*S_(3)" "...(3)` Subtracting (2) from (1), we get `W_(1)-W_(2)=V(S_(2)-S_(1))` Subtracting (3) from (2), we get `W_(2)-W_(3)=V(S_(3)-S_(2))` Subtracting (1) from (3), we get `W_(3)-W_(1)=V(S_(1)-S_(3))` `therefore" "S_(1)(W_(2)-W_(3))+S_(2)(W_(3)-W_(1))+S_(3)(W_(1)-W_(2))` `=S_(1)V(S_(3)-S_(2))+S_(2)V(S_(1)-S_(3))+S_(3)*V(S_(2)-S_(1))` `=V[S_(1)(S_(3)-S_(2))+S_(2)(S_(1)-S_(3))+S_(3)(S_(2)-S_(1))]` = 0.
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