172 



SCIENCE. 



[N. S. Vol. V. No. 109, 



and by transposing 



© <?=!+— v^" 



■which is the triangle of rest. 



Change of velocity : If in the following 

 equation (5) the second term of the first 

 member is given as having changed into the 

 first, then the change of velocity is 



® I ' = 1+- = ^ 



Polygon of velocity: If anj' number of ve- 

 locities are given to be added, 



i+-^+ + 



-^\ 



which is the polygon of velocities, and all 

 possible constructions are equivalent to a 

 mere change in the order of the terms. If 

 we change the sign and direction of the 

 arrow in the second member of (6) and 

 then transpose the term to the first mem- 

 ber 



® 



;+->+|-l-« — +\=6'. 



which is the polygon of rest. 



Acceleration : That accelerations may be 

 compounded like velocities students assert 

 readily enough, but few really understand 

 the assertion. Defining acceleration in the 

 usual way, the product of a time factor and 

 a vector is here encountered. But the time 

 factor is scalar and can be fully given by 

 an ordinary number. Let i! be a sufficiently 

 small interval of time. Then, for the case 

 of linear acceleration, the equation reads 



where the quantity in parenthesis is the ob- 

 served change of velocity in the time t. 

 The result merely calls for an increase in 

 the length of the reduced vector, 1/t times. 

 The more genei-al case corresponding to (5) 

 may be taken at once, whence, 



f(l— )=f(i+-H\) 



may be compounded (using a common time 

 t for brevity), as follows : 



The quantities really compounded are 

 thus the velocities (ultimately displace- 

 ments) and the eSect of the scalar factor is 

 a mere change of the length of the arrow 

 produced. 



The case of a finite acceleration and van- 

 ishing t is particularly remarkable. 



Momentum: If to denote mass, we again 

 have the product of a scalar and a vector, in 

 which, therefore, m is fully given by a num- 

 ber. To compound 



we virtually reproduce (1). If the mo- 

 menta are referred to different masses, as in 



it will be necessary to change the length of 

 each arrow before compounding. The 

 proposition may be extended to the polygon 

 of moments, etc., as already shown. 



Force: If the interval t is sufficiently 

 small, force is defined, in a general way, by 



f(l-'H=f(t+-^=f(/'). 



Two accelerations of the general kind 



where in the first member the second term 

 (vector) is changed to the first term in the 

 time t for each particle of the mass to. The 

 quantity compounded is again the product 

 of a vector (velocity) and a scaler m / t. 

 To compound forces we thus virtually com- 

 pound velocities and increase the length 

 of the arrow resulting m / t times. If two 

 forces actuate m we have in the most gen- 

 eral case 



@ f(i-*-)+f(|— .)=f(\). 



These forces might have been rated in terms 

 of different masses, to and to', and times, t 

 and t'. In such cases the first resultant 

 would be multiplied m / t times and the 



