1868.] Mr. J. C. Maxwell on Governors. 281 



the spring which determines S is also arranged so that 



l=¥'-'"Q. ('^) 



the equation will become, if 2ffh=to^'r^j 



0=.(f-.)+?f.Q(|-.). (.5) 



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

 stant unless the velocity of rotation is w. 



The condition about the overflow is probably difiicult to obtain accurately 

 in practice ; but very good results have been obtained within a con- 

 siderable 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 Umited by a hole, there would be a minimum 

 velocity instead of a maximum. 



The diff'erential equation which determines the nature of small disturbances 

 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 ordinary system of diff'er- 

 ential bevel wheels ; or they may be parallel, as in several contrivances 

 adapted to clockwork. 



Let ^ and 77 represent the angular position about each of these axes re- 

 spectively, that of the main shaft, and that of the governor ; then 

 and are linear functions of 4 and and the motion of any point of the 

 system can be expressed in terms either of ^ and 77 or of and 



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

 «be doo d^ , dr] /,x 



with similar expressions for the other coordinate directions, putting suflixes 

 2 and 3 to denote the values of p and q for these directions. Then La- 

 grange's equation of motion becomes 



sa«+m,-..(Jf-a.+ ga,+ ga.)=o, . . (2) 



where /S? and H are the forces tending to increase ^ and rj respectively, no 

 force being supposed to be applied at any other point. 

 Now putting 



a^=j9, -fg^ (3) 



de ^'de^^'de* ^ ^ 



