ON THE HYPOTHESIS OF UNDULATIONS. 369 



/ is a function of r. And the equation (12) is quite consistent with this assumption. In fact, for 

 this case it becomes, 



f ^ / , , ' 

 ar' r dr f 



wliich equation determines the particular form of f applicable to common light. This equation does 

 not appear to be exactly integrable. By putting it under the form, 



d'f df df 

 dr fdr' rdr 



it will be seen that the equation /= cos -^z- satisfies it, when r is very small. By multiplying 



-/2 



equation (13) by /, and supposing -— = o, we shall have either / = 0, or -j\ + kn-f==0. The 



dr dr~ 



latter equation is satisfied at the axis of the ray : the other by a certain value I of r, which may be 



called the radius of the ray. If S equal the condensation at the axis, and s the condensation at a 



point distant by r from the axis, by what has been already shewn, s = Sf. Hence where r = I, 



(Is 

 both s = 0, and — = ; that is, at this distance there is neither condensation nor variation of con- 

 dr 



densation. Thus the parts of the fluid more distant from the axis than / may remain at rest, while 



those at less distance continue in agitation. As a'^ = o^(l + k), and as it is not probable that a' 



differs much from a, k may be considered a very small numerical quantity. Hence the three first 



terms of equation (13) will be small, since each would vanish if k vanished. Consequently / the 



radius of the ray must be large compared to X the breadth of an undulation. Because —— is very 



small for all values of r, and f and -— vanish together where r = l, it follows that the second term 



of equation (13) is very small compared to the others at all distances from the axis. By neglect- 

 ing this term the equation becomes, 



--^ + -f + kn'f^O (14), 



dr rdr 



which determines with sufficient approximation the function /. 



By neglecting in the general equation (12) the terms containing ~~^ and -r^ , which are 

 quantities of the same order as the neglected term of equation (13), we obtain, 



^.-^^+^«y=0 (.5), 



which is a general equation, applying to a ray of light of any kind, and including as a particular 

 case equation (H). Since by hypothesis s = Sf, we immediately derive from (15), 



d- 8 d' s 



_+__^A„^ = o (ifi), 



a linear equation with constant coefficients, in which the principal variable is tlie condensation. 



The velocity (u) in the direction of .r we find to be (b — , which, since s = Sf, l)ecomes i . 



"^ dx S dx 



Vol. VIII. Pabt III. 3B 



