59 



Theorem * If BD=o, and IG=Qo+Ro-+So'+ &c. Then 

 Resistance (r) : Gravity (g) : : 3S ♦''l+Q- : 4R*. 



Dem. Let f=time of describing CG, then the velocity at C 

 in direction CI=limit of -7- = limit of ;^ X /|^ But Re- 

 sist, in direction CI = lim. '-^ = lim. '"-'""^^^^ Now 

 FG=IG— FI=Ro^ + So3+ &c. Therefore limit ~ = -^ 



dec. 



and dec. -^ =f^ &c. therefore lim. -;^=^. Hence 

 resist, in direct. CI = ^, and consequently whole resis- 

 tance = g X lim. ^^ = §yT^^ and r : ^ : : 3S ^T+^ : 4R«. 

 Q. E. D. 



« Vid. Fig. 



t When CI or o becomes o-|-o', IG becomes Ci(o+o') + R(o-|-o')'+S(o+dy-j- &c. = 



C Hence when AB is increased by o, 



1 (Q+2Ro'+3S«')o Incr. of Q— 2Ro+3So'+ &c. 



1 +(R+3S^+6To')o» Incr. of R=3So+6To^ &c. 



V. &c. &c. &c. &c. 



TJierefore dec. -7= =: -7=. — . =: —= v — &c 



v/R •R v^K+3So+&c. v^R 2R 



I2 



