278 Mr. R. H. M. Bosanquet on Self-regulating 



set off as ordinates p, and the velocities as abscissae v. We 

 shall then obtain a curve corresponding to a relation between 

 p and v, which we may call the governing function of the 

 prime mover or engine. 



When the load is very great it will bring the engine to 

 rest ; then the velocity and power supplied are both zero. 

 Again, when the engine is running free from load it is supply- 

 ing no work, and the velocity has its greatest value. 



The most general expression satisfying these conditions is 

 based upon the form 



p=Tvny.7r, 



where P is the maximum power 

 absorbed, V velocity of free 

 running. 



This law is illustrated in 



%. i. 



By Fourier's theorem any ^=Psin^7r. 



law whatever of this kinol 

 may be represented by an equation of the form 



V V V 



p = F 1 sin ^7r + P 2 sin 2^7r + P 3 sin 3^7r 



+ .... ad infinitum ; 

 where P l7 P 2 , &c. are coefficients, positive or negative, to 

 be determined in accordance with the law to be represented. 

 The sharper the point of the curve, the more considerable the 

 higher terms of the series will be. 



These forms, however, are useless for simple purposes ; and 

 we have to consider the representation of governing relations 

 by simple approximate formulae. 



The simplest assumption for many purposes is that the 

 curve consists of two straight lines, forming a triangle whose 

 base is V, the velocity of free running. The altitude of the 

 vertex is P, the maximum power expended. If we suppose 

 the vertex a little rounded, this may be adjusted to represent 

 many cases. Fig. 2. 



Along OP the law may 

 be written p = av. 



Along PV it may be 

 written p= — /3(v — V). 



The latter assumption 

 is not convenient for our 

 present purpose ; and we 

 shall prefer to represent 

 the falling part of the curve, which shows the cutting-off of the 



p== -P(v-V). 



