158 CENTRE OP G RAVI I 



v of gravity; r lt r t , ... the distances of tin particles 

 the fixed point ; then 



r, f = a 1 + F + 7* - 2 (or, + fa + 7*,) + p,', 



Multiply these equations by m,, m t , TO,,... respectively, and 

 add; then 



- (a* + P + 7 s ) Sm - 2 (aSmx + #5roy + ySm^) + Snip*. 

 But, since the origin is the centre of gravity of the system, 



'Zmx = 0, Smy = 0, 'Zmz = 0, 

 therefore Swir* = (a f + /8" + 7*) 2m 4- 2mp 2 . 



Now Swtp* is independent of the position of the givr-n 

 point; hence the least value of 2mr* is that which it lias 

 when a s + f? + 7* vanishes, that is, when the given point is 

 at the centre of gravity of the system. 



139. Let crj, /9,, 7 t , be the angles wliich p l makes with tlie 

 axes; S , 8 , 7 2 , the angles which p 2 inakrs with the ; 

 and so on ; then we have, supposing the origin the centre of 

 gravity of the system, 



2,mp cos a = 0, *mp cos = 0, 2m/> cos 7 = 0. 



Square each of these equations and add the results; then if 

 7H, m' represent any two masses, and (o, p) the angle between 

 the straight lines which join them with the centre of gravity, 



SroV + 22mm'pp' cos (p, p) = 0. 

 But 2pp' cos (p, p') = p* + p' f - w*, 



where u denotes the distance of TO and TO'. Hence 



STO'P' + STOTO' (p 1 + p" - w*) = 0. 

 If we select the coefficient of p t f , we find it to be 

 TO,* + TO, (m t +tfi, + ...) orw.: 



and the other coefficients are similar. Hence the above 

 equation may be written 



