MINDING’S SYSTEM OF FORCES. 523 
Second Method. 
Formule connecting the co-ordinates of the foot of the perpendicular on a 
central axis or single resultant. 
MR-—N 
if g= ee = Giz Agthr,.rg . E ‘ (1) 
7 = SO = GIy pg thy. ps ; ; (2) 
= S&C, = Op, . Vo + hr. 3 ; (3) 
in terms of former notation, the equation to the central axis is 
gl ANE a 
Ay hy My? 
and it is easy to see that €)¢ are the co-ordinates of the foot of the perpen- 
dicular from the origin upon it. 
If p denotes the length of this perpendicular, we get by squaring (1) (2) (3) 
and adding, 
P= Op, Phy -. ; ‘ ; ; (5) 
If the central axis be a line of action of a single resultant, we have as we 
saw before 
Ip; = hv, : : : : : : (6) 
Using (6) we get at once from (5) (2) (3) 
PHP Py th’, 4 
IN=F by» Ma th’ry . vs 
GL=G by - Pg thr, vs 
whence 
Pt+GV7tVE=gipyth . : (7) 
The equations (5) and (7) form the basis of the following discussion ; (5) holds 
for all central axes, and is the onefold relation that determines the complex 
which they form; (5) and (7) taken together give the twofold relation which 
determines the congruency of single resultants. 
In what follows we shall replace the equations (5) and (7) by 
PHP AY +9 ey +h’ (8) 
Pt+PEt Git VCH=fr~t+y' pth,’ . ; (9) 
The formule thus found will be more symmetrical, and MINDING’s case is 
obtained by putting f=0. 
