THEORY OF VIBRATIONS 53 



in ascending order of frequency ; the mode of slowest vibration 

 may still be called the " fundamental," and is generally the 

 most important. 



Before leaving the general theory it may be desirable to 

 emphasize once more the importance of the simple-harmonic 

 type of vibration from the dynamical point of view. We have 

 seen that it is the characteristic type for a frictionless system of one 

 degree of freedom, or (more generally) for a system oscillating 

 as if it possessed only one degree, as in the case of the normal 

 modes. It is also the only type of imposed vibration which is 

 accurately reproduced, on a larger or smaller scale, in every 

 part of the system. If a force of perfectly arbitrary type act at 

 any point, the vibrations produced in other parts of the system 

 have as a rule no special resemblance to this or to one another; 

 it is only in the case of a periodic force following the simple- 

 harmonic law of variation with the time that the induced 

 vibrations are exactly similar, and keep step with the force. 

 Moreover it is only in so far as the disturbing force is simple- 

 harmonic, or contains simple-harmonic constituents, that it is 

 capable of generating a forced vibration of abnormal amplitude 

 when a critical frequency is approached. It is in these circum- 

 stances that Helmholtz found the clue to his theory of audition, 

 to which we shall have to refer at a later stage. 



20. On the Use of Imaginary Quantities. 



The treatment of dynamical equations can often be greatly 

 simplified by the use of so-called "imaginaries." As we shall 

 occasionally have recourse to this procedure, it may be convenient 

 to explain briefly the principles on which it rests. 



The reader will be familiar with the geometrical representa- 

 tion of a "complex" quantity a+ ib, where a, b are real and i 

 stands for \/( I), by a vector drawn from the origin to the 

 point whose rectangular coordinates are (a, b), and with the 

 fact that addition of imaginaries corresponds to geometrical 

 addition (or composition) of the respective vectors. The 

 symbol a + ib when applied as a multiplying operator to any 

 vector denotes the same process by which the vector a + ib may 

 be supposed to have been derived from the vector 1, viz. it 



