396 Royal Society : — 



If the conditions of overflow can be so arranged that the mean 



Q2 



square of the velocity s represented by — , is proportional to Q, and if 

 the strength of the spring which determines S is also arranged so that 



*■#** ("> 



the equation will become, if 2gh = io q r' 2 , 



°=<f -" )+»^ -•)'■ C15) 



which shows that the velocity of rotation and of overflow cannot be 

 constant unless the velocity of rotation is w. 



The condition about the overflow is probably difficult to obtain accu- 

 rately in practice ; but very good results have been obtained within 

 a considerable range of driving-power by a proper adjustment of the 

 spring. If the rim is uniform, there will be a maximum velocity for 

 a certain driving-power. This seems to be verified by the results 

 given at p. 667 of Mr. Siemens' s paper. 



If the flow of the fluid were limited by a hole, there would be a 

 minimum velocity instead of a maximum. 



The differential equation which determines the nature of small dis- 

 turbances is in general of the fourth order, but may be reduced to 

 the third by a proper choice of the value of the mean overflow. 



Theory of Differential Gearing. 



In some contrivances the main shaft is connected with the governor 

 by a wheel or system of wheels which are capable of rotation round 

 an axis, which is itself also capable of rotation about the axis of 

 the main shaft. These two axes may be at right angles, as in the or- 

 dinary system of differential bevel wheels ; or they may be parallel, 

 as in several contrivances adapted to clockwork. 



Let £ and rj represent the angular position about each of these 

 axes respectively, that of the main shaft, and that of the go- 

 vernor ; then 6 and </> are linear functions of £and rj, and the motion 

 of any point of the system can be expressed in terms either of £ and 

 r) or of 6 and <p. 



Let the velocity of a particle whose mass is m resolved in the direc- 

 tion of x be 



dec, dt, . dr) /1N 



iT*3 + *W (1 > 



with similar expressions for the other coordinate directions, putting 

 suffixes 2 and 3 to denote the values of p and q for these directions. 

 Then Lagrange's equation of motion becomes 



