330 Prof. Challis on the Undulatory Theory of Light. 

 and we may assume/ to be such that, where x = and ?/ = 0, 



A> 2=°< 1=0. 



The details of the reasoning here referred to are given in the 

 proof of Proposition X. contained in an Article on the Princi- 

 ples of Hydrodynamics in the Philosophical Magazine for 

 December 1852. In the same Article are investigated exact 

 expressions for the functions / and <j> 3 and the rate of propaga- 



/ el& 



tion of the motion along the axis is found to be a\ / 1+ —^- 3 



a and \ having the usual significations, and e being a constant 

 such that, where #=0 and y=0, 



_/ + _/ +<k _0. 

 dx 2 dy 2 



As e is necessarily a positive quantity, it follows that the rate of 

 propagation, as determined by hydrodynamics, is greater than a. 

 It is further evident, putting /ca for the rate, that if k be a nu- 

 merical constant, its value should be determinable exclusively on 

 hydrodynamical principles. This is what I have attempted to 

 do in a communication to the Philosophical Magazine for Fe- 

 bruary 1853 ; but having recently discovered that the mathema- 

 tical reasoning there given requires correction, I propose now 

 to enter upon the discussion of this point. 



The determination of the constant e depends on the integra- 

 tion of the equation 



dr 2 rdr ' 



r being any distance from the axis of motion. The integral is 

 not obtainable in a finite form, but it may readily be shown that 

 the following series for /satisfies it, viz. 



e 2 r 4 e 3 r 6 



/ = l_ er 2 + ________ +&c< 



For finding e it is required to ascertain the large values of r that 

 make/ vanish. This problem is solved by Sir W. Hamilton in 

 a memoir on Fluctuating Functions in the Transactions of the 

 Royal Irish Academy (vol. xix. p. 313), and by Professor Stokes 

 in the Transactions of the Cambridge Philosophical Society 

 (vol. ix. part 1. p. _82). I hesitated to accept the equation (52) 

 in the latter memoir, because it contains quantities R and S 

 representing series that are convergent for a certain number of 

 terms and then become divergent, which yet are employed as if 

 they were wholly convergent. Whatever be the answer to this 



