334 Prof. Challis on the Undulatory Theory of Light. 



There remains another consideration which must be brought 

 to bear on the determination of the velocity of propagation. 



We found above the general relation w = — between the Ion- 



gitudinal velocity and condensation in a single series of waves. 

 The analogous relation obtained by the process of reasoning 

 that has been usually adopted in questions of this kind is w=-aa. 

 But that process does not take into account that the total motion 

 is composed of separate longitudinal and transverse motions 



relative to axes. The factor - is wholly due to the lateral 



K " 



spreading which accompanies the condensations and rarefactions 



propagated along and parallel to the axis of motion, which has 



the effect of diminishing the rate of change of the density in the 



direction of propagation, and thus making the effective elasticity 



of the fluid, ceteris paribus, less than the actual in the ratio of 



d z 



-g to a 2 . But just in the proportion in which the effective 



elasticity is caused by lateral spreading to be less than the actual 

 in the direction of propagation, it must, by a reciprocal action, 

 be made greater than the actual in the transverse direction, and 

 accordingly be increased in the ratio of /c 2 a 2 to a 2 . Thus the 

 ratio of the latter effective elasticity to the other is k 4 , and the 

 ratio of the corresponding velocities of propagation is a; 2 . Now 



we have proved that this ratio is r-. Hence, substituting in the 

 expression for tc } we have 



*=\/' 1+ ;? 



or /c 6 — k 4 = 1. 



Consequently the numerical value of k 1 is obtained by the solu- 

 tion of a cubic equation which has one real positive root and two 

 imaginary roots. The value of tc will be found to be 1*2106. 

 Hence, taking a = 916*322 feet, the resulting velocity of propaga- 

 tion is 1109*3 feet. The value by observation, as given by Sir 

 J. Herschel in the Encyclopedia Metropolitana, is 1089*7 feet. 

 The difference 19*6 feet might be lessened in some degree by 

 calculating the corrections of the observations for temperature 

 according to Regnault's coefficient of expansion. But probably 

 the principal part of the difference is due to the circumstance 

 that the theoretical reasoning assumes the fluid to be perfect, 

 andit may be that atmospheric air is not strictly such. It seems 

 hardly to be accounted for that a course of reasoning involving 

 considerations so various and peculiar as those which have been 

 gone through above, should have conducted to a result differing 



