Page:On Governors.pdf/11

Rh Let the velocity of a particle whose mass is $$\scriptstyle\ m$$ resolved in the direction of $$\scriptstyle\ x$$ be

with similar expressions for the other coordinate directions, putting suffixes 2 and 3 to denote the values of $$\scriptstyle\ p$$ and $$\scriptstyle\ q$$ for these directions. Then Lagrange's equation of motion becomes

where $$\scriptstyle\ \Xi$$ and $$\scriptstyle\ H$$ are the forces tending to increase $$\scriptstyle\ \xi$$ and $$\scriptstyle\ \eta$$ respectively, no force being supposed to be applied at any other point.

Now putting

and

the equation becomes

and since $$\scriptstyle\ \delta x$$ and $$\scriptstyle\ \delta \eta$$ are independent, the coefficient of each must be zero.

If we now put

where $$\scriptstyle\ p^2 = p_1^2+p_2^2+p_3^2, \ pq = p_1 q_1+p_2 q_2+p_3 q_3$$, and $$\scriptstyle\ q^2 = q_1^2+q_2^2+q_3^2$$, the equations of motion will be

If the apparatus is so arranged that $$\scriptstyle\ M = 0 $$, then the two motions will be independent of each other; and the motions indicated by $$\scriptstyle\ \xi$$ and $$\scriptstyle\ \eta$$ will be about conjugate axes—that is, about axes such that the rotation round one of them does not tend to produce a force about the other.

Now let $$\scriptstyle\ \Theta$$ be the driving-power of the shaft on the differential system, and $$\scriptstyle\ \Phi$$ that of the differential system on the governor; then the equation of motion becomes

and if

and if we put