224 On Equilibrium and Initial and Steady Motions. 



expressed by the vanishing of the first r coordinates yfr x . . . ^ r . 

 The relation of the two cases to be compared is expressed 

 by supposing the forces of the remaining types ^F r +i, ... to 

 be the same, so that A^+j &c. vanish. Thus for every suffix 

 either yfr vanishes or else A"^. Accordingly S^JrA^ is zero, 

 and therefore also, by the law of reciprocity, S^A^Jr. Hence, 

 as above, 



2AV=2A^A>|r, (15) 



showing that the removal of the constraint increases the poten- 

 tial energy by the potential energy of the difference of the de- 

 formations. 



Corresponding to the above theorems for T and V, there are 

 two more relating to the function F introduced by me in a 

 paper printed in the * Proceedings of the Mathematical Society ' 

 for June 1873, expressing the effects of viscosity. We have 

 here to consider systems destitute both of kinetic and poten- 

 tial energy, of which probably the best example is a combi- 

 nation of electrical conductors, conveying currents, whose in- 

 ductive effects, dependent on inertia, may be neglected. The 

 equations giving the magnitudes of the steady currents are of 

 the form 



^=% (16) 



dyjr 



where F is a quadratic function (in this case with constant co- 

 efficients) of the velocities -^, &c, representing half the dissipa- 

 tion of energy in the unit of time, and ^ &c. are the electro- 

 motive forces. It is scarcely necessary to go through the proofs, 

 as they are precisely similar to those already given with the sub- 

 stitution of F for T, and steady forces for impulses. The ana- 

 logue of Bertrand's theorem tells us that, if given electromotive 

 forces act, the development of heat in unit time is diminished 

 by the introduction of any constraint, as, for example, breaking 

 one of the contacts. And by comparison with Thomson's theo- 

 rem for initial motions we learn that, if given currents be main- 

 tained in the system by forces of corresponding types, the 

 whole development of heat is the least possible under the cir- 

 cumstances (Maxwell's 'Electricity and Magnetism/ § 284). 

 And precisely as before, we might deduce corollaries relating 

 to the effect of altering the resistance of any part of the 

 combination. 



