190 IOWA ACADEMY OF SCIENCE 



Turning now to the arrangement shown in diagram 3 we ar- 

 rive, by the two methods of approach to equations identical 

 with equations (2) and (3), respectively. However, in this case, 

 m is less than unity, so that equation (2) leads to a negative 

 value for /, which means that no positive pull will yield equi- 

 librium, and hence that the physical solution is impossible. The 

 discussion of the two cases when m equals unity is obvious. The 

 scries (3) becomes now a divergent series, and cannot be sum- 

 med. Here then the second method of approach fails to yield 

 any result, whereas the algebraic method does yield a result 

 although it has no physical reality. A glance at the divergent, 

 series (3) shows that the man is forced to exert a greater and 

 greater pull, which situation would no doubt correspond with 

 the facts in an actual situation. But the more the man pulls the 

 more certain he is of falling to the ground. 



The most interesting case, how T ever, is the last one, repre- 

 sented in diagram i. Here we arrive by the two methods of 

 approach at equations (6) and (7), respectively. The alge- 

 braic solution (6), is perfectly definite and physically possible, 

 even when m is less than unity. However, the series (7) is di- 

 vergent, when m is less than unity, and ordinarily considered 

 it has no sum. A glance at the series will show that the man 

 first pulls with a certain force, then relaxes the tension by a 

 greater amount, next pulls with a still greater force, and so on, 

 pulling and relaxing with forces ever increasing in magnitude. 

 It is evident that this latter method would not be the actual one. 

 and it again becomes a matter of interesting speculation as to 

 with what rhythmical, or other, muscular efforts the man arrives 

 at the correct force for equilibrium. It is possible that the terms 

 of the divergent series (7) could be grouped in a certain fashion 

 to yield a convergent series which would have the correct sum. 



It is evident that the second method of analysis of the prob- 

 lem succeeds in cases shown in diagrams 1 and 2, for all values 

 of m greater than unity, fails in 3, for m is less than unity, 

 but fails because the solution is impossible, and fails utterly in 

 4 for values of m less than unity, although the solution of the 

 problem is possible and perfectly definite. 



Physics Laboratorv, 

 State University op Iowa. 



