90 -. ir.u. .v 



We proceed to apply these equations to find the least 



value ul ' 



To find the locus of the origins which give the least prnx-ifnl 



i cuts, the magnitude of those moments, and (/> 

 their axes. 



:ijilv the first of equations (1) by A', the second by 

 1 ) . in 1 the third by *Z, and add; thus 



/.* A'-f J/'S r+ .V'2^= LSX+ J/2 r+ NZZ... (2). 

 Also 



= {(SA r ) f + (2 F) -f (2Z)'J {// 



- M'ZZ)* + (Z/2Z- J^l 



Of these four terms the last is constant for all valu 

 x' t ;/', ^' by (2) ; hence we obtain the least value of G' by 

 making the three preceding terms vanish, which gives 



m 



r " 



that is, 



L-i/'2Z+t'2Y M-ztX+x'2Z N-x'ZY+i/'ZX 

 ~2Y~ ^) ~^^ 



c the required locus is a straight line. 

 From (4) it appears that L\ M\ N' are proportional to 



. i Y y 2Z respectively, which shews that the axis of the 

 principal moment at any point on the straight line (5) is 

 parallel to the direction of tne resultant JR. By (3) the value 



the least principal moment is 



LZX+MZY+NZZ 



~ir 



h of the fractions in (5) is, by a known theorem, 

 equal to 



that is, to 



It 1 



