On the Self-induction of Wires. £ 333 



Certainly ; but there is much more in it than that. For 

 not only the conjugate properties of spherical harmonics, but 

 those of all other functions of the fluctuating character, which 

 present themselves in physical problems, including the infi- 

 nitely undiscoverable, are involved in the principle of energy, 

 and are most simply and immediately proved by it, and pre- 

 dicted beforehand. We may indeed get rid of the principle 

 of energy, and treat the matter as a question of the properties 

 of quadratic functions ; a method which may commend itself 

 to the pure mathematician. But by the use of the principle 

 of energy, and assisted by the physical ideas involved, we are 

 enabled to go straight to the mark at once, and avoid the un- 

 necessary complexities connected with the use of the special 

 functions in question, which may be so great as to wholly 

 prevent the recognition of the properties which, through the 

 principle of energy, are necessitated. 



Considering only a dynamical system in which the forces of 

 reaction are proportional to displacements, and the forces of 

 resistance to velocities, there are three important quantities — 

 the potential energy, the kinetic energy, and the dissipativity, 

 say U, T, and Q, which are quadratic functions of the variables 

 or their velocities. When there is no kinetic energy, the 

 conjugate properties of normal systems are U 12 = and Qi 2 = 0; 

 these standing for the mutual potential energy and the mutual 

 dissipativity of a pair of normal systems. When there is no 

 potential energy, we have T 12 = and Q 12 = 0. When there 

 is no dissipation of energy, U 12 = and T 12 = 0. And in 

 general, U 12 = T 12 , which covers all cases, and has two equiva- 

 lents, -J Qi2 + tJi2 = 0? and iQi 2 + T 12 = 0; for, as the mutual 

 potential and kinetic energies are equal, the mutual dissipa- 

 tivity is derived half from each. 



Let the variables be # 1? «%, . . . , their velocities v 1 = A' 1 , . . . , 

 and the equations of motion 



F 1 = (A 11 + B 11 p + C 11 /> 1 + (A 12 + B 12 p + C 12 p> 2 + ..., ^ 



F 2 = (A 21 + B 21 ^ + C 21 /K+(A 22 + B 22 p + C 22 p> 2 + ..., > (88) 



where F 1? F 2 , . . . , are impressed forces, and p stands for d/dt. 

 Forming the equation of total activity we obtain 



where 



2F^ = Q + U + T; (89) 



>x v v 2 + A 22 x 2 2 + ... ) ^ 



\V 2 + C 2 2 v l + '-' 'J 



2U = k. n x\ + 2 K 12 x x x 2 + ^22^1 + • 

 Q = B n ^+2B 12 ^ 2 + B 22 i^+... 

 2T=C n ^ + 2( V, 



