422 Mr. C. W. Merrificld on the Problem of Rectilinear Motion. 



practical convenience. The resulting curves we will call OU , 

 OUj, and OU 2 respectively. 



The ordinates of these new curves will be the spaces traversed. 



Our problem is now solved. Along the left-hand side of our 

 sheet we have t as the current abscissa or equidistant variable, 

 and we have the ordinates of the curves 



OW , OW„ OW 2 for the velocities, 



OU , OUj, OU 2 for the spaces. 

 It is to be observed that the curve OW„ giving the rising velo- 

 city, corresponds to -=-§, and the curve OW 2 , the falling 



j j i 

 velocity, to 



1 _1 



y-kv 3 g 



As an illustration of the use of the diagram, let us suppose 

 that a body is projected upwards with a given velocity, and that 

 we require to compare its total time of flight with what it would 

 have been in vacuo. 



Set off v } along OV. Draw the ordinate v 1 p 1 to meet the 

 curve OW x . Draw p x q l to meet the curve OUj parallel to OV. 

 Now the space of fall has to be equated to the space of rise. 

 Hence we draw, to meet the curve OU 2 in q 2 , a right line q Y q^ 

 parallel to OT. The distance of q 2 from OV gives the time ; and 

 by drawing q 2 p 2 to meet OW 2 in p 2 we get the velocity at the 

 lowest point. We thus get the times, spaces, and velocities in the 

 resisting medium, and we have only to perform similar opera- 

 tions on the vacuum-lines to complete the comparison. In fact 

 if Pq be the point corresponding to v on the curve OW , the dif- 

 ference of time in the resisting medium is given at once by 



2v l p -{v 1 p l + v 2 p 2 ); 



and it is easily seen that we may obtain, by simple constructions, 

 any other such comparison as we may desire. 



Besides being convenient for the purpose of comparison, the 

 curves for vacuum enable us to verify our units of scale. This is 

 a point at which error is very likely to creep in. 



It has been suggested to me by some friends, to whom I have 

 shown this, that it might be desirable to publish it, as an example 

 of the value of graphical methods, and also as the graphical so- 

 lution of a question to which considerable interest now attaches, 

 with reference to the theory of projectiles. The chief point of 

 novelty is the change of the independent variable by the inver- 

 sion of the diagram. 



