GOVERNORS. 119 



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

 tively, that of the main shaft, and < that of the governor; then 6 and <f> 

 are linear functions of and 77, and the motion of any point of the system can 

 be expressed in terms either of and 77 or of 6 and <f>. 



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



x be 



dx d 



with similar expressions for the other co-ordinate directions, putting suffixes 

 2 and 3 to denote the values of p and q for these directions. Then Lagrange's 

 equation of motion becomes 



where H and H are the forces tending to increase f and 77 respectively, no 

 force being supposed to be applied at any other point. 



Now putting Sx=p 1 8t; + q 1 T), ................................. (3), 



and 



the equation becomes 



and since 8 and 877 are independent, the coefficient of each must be zero. 



If we now put 



2 (rap 5 ) = L, 2 (inpq) = M, S (rag" 2 ) = N (6), 



the equations of motion will be 



If the apparatus is so arranged that M=0, then the two motions will be 

 independent of each other ; and the motions indicated by f and -r\ will be about 



