134 THE ROYAL SOCIETY OF CANADA 



R2 R3 



If the equivalent resistance of R2 and R3, namely — — : — — be 



R2 + R3, 



represented by r then it can easily be shown that 



V^ / 1 1 



^' 4.18 VRi + r Ri + R2/ ^^^ 



The temperature variation during regulation, expressed in terms 

 of these known or easily ascertained quantities, is therefore approxi- 

 matelv 



4.18 (pi + P2) MS \ Ri + r Ri + R2/ 

 degrees centigrade: 



In designing alterations, or in searching for optimum conditions 

 it is useful to note that 



- v-^ 



^- ~ 4.18 (Ri + Ro) ^^' 



and from equation (1) we see that 



Pi 

 Hi = H2 + , H3 



Pi + P2 



and substituting in this, the values of H2 and H3 given in equations 

 (4) and (2) we get 



V- / P2 Pi \ 



^^ " 4.18 (pi + P2) U1+R2 "^ Ri + rj ^^^ 



If Hi is less than 25 calories per second and alters by only a small 

 percentage per degree change in external temperature, one can choose 

 values which give very small ranges of variation. For example 

 consider the following sample cases for our thermostat regulating at 

 25-00°C, while the external temperature is falling. 

 Case 1. 



In each case 

 External temperature 19-5°C pi = 55 seconds V = 110 volts 



P2 = 150 " M = 45,000 grams 



S = 0.52 

 Range of variation by equation {3) = 0.0049° C Ri = 75 ohms 



R2 = 75 ohms 

 and by equation (5), Hi = 20.1 calories per R3 = 220 ohms 

 sec. (r = 56 ohms) 



Case II. (H2 = 19.3 cal /sec.) 



(H3 = 2.85 " " ) 

 External temperature 17.5°C pi = 75 seconds. 



po = 85 " 

 Range of variations, by equation {3) = 0.0047°C. 

 and by equation (5) Hi = 20.6 calories per second. 



