30 



Mating Q and 7?, we shall obtain 



Pss 8 *4fy^t-yO+y4 (**-**) + ^foyt-- r 



for the vnluim- 



'irivcn in works on Analytical (imm-try of 

 visions, we have 



Pj_ volume of pyramid OBCD 

 5~~ volume of pyramid OABC' 



f) /' 



Similarly we obtain the value of ' and of -~. 



EXAMPLES, 



1. T\vo forces 1 Q have a resultant .// which n. 



nn angle a with P be increased by 7i' whil.- V remains 



unchanged, shew that the new rc.sultant makes an angle 



fhv. 



2. Two forces in the ratio of 2 to \/3- 1. an in< lit, 



an ande of 60; what must be the lin < tiou aiid 

 magnitude of a third force which produces equilibrium? 



Jfc> v required force must be to the first of t 



forces as and its direction produced makes an angle 



with that force. 



3. The resultant of two forces P and Q is equal to 

 and makes an angle of 30 with J'\ find P in terms . 



Result. P= Q or P=2#; in the former case the 



angle between P and Q i* 60, in the latter 1 



/), E t -Fbe the middle points of the sidos of the 

 triangle ABC and any other point, shew that t! 

 of forces represented by OD, OE, OF is equivalci 

 represented by OA, OB, OC. 



