xxi.J THEORY OF APPROXIMATION. 469 



This hypothesis happened to be approximately true in 

 the case of air, so that no error was discovered in ex- 

 periments on sound. Had it not been so, the earlier 

 analysts would probably have failed to give any solution, 

 and the progress of the subject might have been retarded. 

 Cauchy was able to make a new approximation under 

 the more difficult supposition, that the particles of the 

 vibrating medium are situated at, considerable distances, 

 and act and react upon the neighbouring particles by 

 attractive and repulsive forces. To calculate the rate of 

 propagation of disturbance in such a medium is a work 

 of excessive difficulty. The complete solution of the 

 problem appears indeed to be beyond human power, so 

 that we must be content, as in the case of the planetary 

 motions, to look forward to successive approximations. 

 All that Cauchy could do was to show that certain quan- 

 tities, neglected in previous theories, became of consider- 

 able amount under the new conditions of the problem, 

 so that there will exist a relation between the length of 

 the wave, and the velocity at which it travels. To re- 

 move, then, the difficulties in the way of the undulatory 

 theory of light, a new approach to probable conditions 

 was needed. 1 



In a similar manner Fourier's theory of the conduction 

 and radiation of heat was based upon the hypothesis that 

 the quantity of heat passing along any line is simply pro- 

 portional to the rate of change of temperature. But it 

 has since been shown by Forbes that the conductivity of a 

 body diminishes as its temperature increases. All the 

 details of Fourier's solution therefore require modification, 

 and the results are in the meantime to be regarded as 

 only approximately true. 2 



We ought to distinguish between those problems which 

 are physically and those which are merely mathematically 

 incomplete. In the latter case the physical law is cor- 

 rectly seized, but the mathematician neglects, or is more 

 often unable to follow out the law in all its results. The 

 law of gravitation and the principles of harmonic or un- 

 dulatory movement, even supposing the data to be correct, 



1 Lloyd's Lectures on the Wave Theory, pp. 22, 23. 

 a Tait's Thermodynamics, p. 10. 



