FUNDAMENTAI PRINCIPLES. 17 
points in value respectively 1, 2, 3, 4, 5; required the 
center of gravity of the system. 
Take any two points, as2 and 3, and find their mutual 
center of gravity, just as in Fig. 9. This will be 
located at a point at which the whole value, 5, of the 
2-3 system may be assumed to be concentrated. Now 
find the center of gravity of this point and 5; this will 
be at a point at which the weight will be 10. “Then 
taking 1 and 4, the resultant weight will be 5. Finally, 
balance these resultants and the center of gravity of the 
entire system is found at 15. ‘The order in which the 
combinations are made is of no consequence, since a given 
system can have but one center of gravity. Now, suppose 
the points 1, 2, 3, 4, 5, are supplied from a common source 
O, Fig: 11, through lines' /7y, /,,4,,/,, /. Refersing to 
equation (5) the total weight of copper in any line, as b 
may be written W ==K ¢/, where K depends on the uniform 
drop assumed. For any number of load points thus con- 
nected to a center O 2 W=XK 2> ¢/2. But thisis directly 
proportional to the moment of inertia, = 7 72, of the loads 
considered as weights, about O as an axis. Now the 
moment of inertia of any body about any axis is composed 
of the sum of two terms, viz., first, the moment of inertia 
of the parts of the body around its center of inertia and, 
second, the moment of inertia of the whole mass concen- 
trated at its center of inertia, about the axis chosen. 
Therefore, the minimum moment of inertia for a given set 
of loads is obtained when the axis coincides with the 
center of inertia, thereby causing the second term to dis- 
appear. Hence the ‘total weight of copper required for 
supplying, at a given loss, any system of loads is a mini- 
mum when the system is fed from its center of gravity. 
And the penalty for disregarding this law is severe, as will 
presently be shown. 
For example, take the case of a circular area with an 
electric system made up of equally and uniformly loaded 
lines radiating from a power station at the center. It has 
already been shown that the cross section of copper needed