49, 50J EFFECT OF LINEAR TERMS. 135 



Using these values in 168), we obtain for the force, 



= ^ sl &' -f ^2 &' + - + #.mgm + -fci 



where the 6r's are functions of the coordinates defined hy 



For the force applied to change a coordinate q t we have a similar 

 form, with coefficients such that 



IT*) . ' *,.-- *.,. 



We have then the result that the terms linear in the velocities in 

 the kinetic potential give rise to reactions linear in the velocities, 

 with the property that the coefficient of q} in the reaction P s is 

 equal and of opposite sign to the coefficient of qj in the reaction P t . 

 Such reactions are called gyroscopic forces by Thomson and Tait 1 ), 

 since we have examples of them where gyrostats, or symmetrical 

 bodies spinning about axes attached to parts of systems, act as 

 concealed cyclic motions. If we find the activity of the gyroscopic 

 forces, 



173 ) d - = 



we find that in the part P S W q s ' we have the term G- st q s ' <Lt while in 

 the part P t q t ' we have the term G- ts q s ' q t ' } and since G ts = G- st j 

 these two terms destroy each other. Accordingly the gyrostatic 

 ^orces_jdisapp-ear _ fc>m. the equation of activity. These forces are 

 consequently conservative motional forces. They are however perfectly 

 distinguishable by their effects from the conservative motional forces 

 arising from the term C which imitates potential energy, and they 

 in no wise imitate potential energy, as we shall see by an example. 

 A system containing gyrostatic members behaves in such a peculiar 

 manner that their presence is easily inferred. The theory of gyro- 

 stats will be treated in Chapter VII. In the mean time the following 

 simple example will illustrate the theory, and at the same time serve 

 to prepare for the general theory of the gyrostat, of which it con- 

 stitutes a special case. 



1) Thomson and Tait, Nat. Phil. 345^1. 



