REVIEW OF CONTROLS OF CAPACITY. 



189 



.(98) 



ments of equation (91), they render probable 

 the general proposition : 



The rate at which capacity varies 

 inversely with each of the three 

 controlling conditions, slope, dis- 

 charge, and fineness, itself varies 

 inversely with each of the condi- 

 tions. 



Returning to (92) and (93), we may indicate 

 certain corollaries. 



Starting from the status of competence, let 

 us assume that slope is increased, with destruc- 

 tion of the status, and that the status is re- 

 stored by reducing discharge. In the restored 

 status IT is greater than in the original, K is less, 

 and <j> is unchanged. It is evident that the 

 nature of the result does not depend on the 

 particular assumptions, and that we may pass 

 to the general proposition: 



When capacity is zero, the compe- 

 tence constants are so related 



that a change in any one of them } (99) 



involves a change of contrary 

 sign in some other. 



Starting from a status characterized by a 

 particular value of capacity, we may first 

 break it by increasing slope and then restore it 

 by decreasing discharge (fineness remaining 

 unchanged). The first change reduces K and 

 <; the second increases a and <. It does not 

 appear whether the net result for (j> involves 

 change in its value, but if so the change is 

 probably small in relation to the increase in a 

 and the decrease in . It is evident that the 

 nature of the result does not depend on the 

 particular assumption, and that we may pass 

 to two general propositions, each of which 

 includes (99) as a special case: 



Under the condition that capac- 

 ity is constant, the competence 

 constants are so related that a 

 change in any one of them in- 

 volves a change of contrary sign 

 in some other. 



Under the condition that capacit - 

 is constant, the values of slope, 

 discharge and fineness are so 

 related that a change in any one 

 of them involves a change of 

 contrary sign in some other. 



..(100) 



(101) 



.(102) 



It follows also that 



Under the condition that capacity 

 is constant, the value of each 

 controlling condition (S, Q, or 

 F) is so related to the corre- 

 sponding competence constant 

 (a, K, or <f>) that the two vary in 

 same sense. 



Propositions (100) and (102) are deduced 

 from equations (93). By parity of reasoning 

 equations (94) yield (103) and (104), but these 

 two propositions share whatever uncertainty 

 attaches to (94). 



Under the condition that capacity 

 is constant, the exponents n, o, 

 p are so related that a change in 

 any one of them involves a 

 change of contrary sign in some 

 other. 



Under the condition that capacity 

 is constant, the value of each 

 controlling condition (S, Q, or 

 F) is so related to the corre- 

 sponding exponent (n, o, or p) 

 that the two vary in the same 

 sense. 



..(103) 



(104) 



As capacity can not be increased under (91) 

 without increasing S, Q, or F, and as the 

 increase of one of these involves under (93) the 

 decrease of two competence constants, without 

 any change of the third, it follows that the 

 competence constants, collectively, vary in- 

 inversely with capacity. The same reasoning, 

 if applied to (91) and (94), yields a similar 

 conclusion as to the exponents. To combine 

 the two in a single statement: 



The competence constants a, K, 

 and </>, taken as a group, and the 

 exponents n, o, and p, taken as 

 a group, vary inversely with 

 capacity. 



I find it not easy to bring into combination 

 the laws of internal relation between para- 

 meters of a group and the laws which connect 

 the groups with capacity; but if these laws be 

 regarded as conditions, it is possible to frame 

 more comprehensive theorems of tentative 

 character. Equations (106) are of this class 



