﻿282 Prof. Challis on Attraction caused 



in the last term for \-rrds its first approximation -J-, all non- 

 periodic terms of the second order will be contained in the ex- 

 pression 



_V* 1 d<j>* 

 2 + 2a' 2 df- ' 

 The finding of such terms requires, therefore, a previous deter- 

 mination of the values of V and -^-. I proceed to do this in a 



simple instance. 



It will be supposed that the vibrations propagated in the fluid 

 are functions of the distance from a fixed point, being continu- 

 ously impressed at a given small distance from the point equally 

 in all directions. Although these circumstances differ greatly 

 from those of the vibrations in Mr. Guthrie's experiments, which 

 were excited by a tuning-fork, the discussion of this case, which 

 has been selected on account of the simplicity of the requisite 

 analytical calculation, may serve to indicate how the " approach" 

 was caused in the instances of those experiments, and generally 

 how a motion of translation can result from the action of aerial 

 vibrations. 



The distance from the fixed point being called r, the solution 

 of the known equation applicable to this case, viz. 



d Q . r<p , 2 d 2 . r<£> 

 ~aW~~ a ~d^~ } 



gives, on the supposition that the vibrations are propagated from 

 the centre, 



Y= f'(r-a't + c) /(r-a't + c) ^ 



r r 2 



, = f'{r-a!t + c) = _d$ m 



r a'dt 



Giving now to the function /the form —m cos --(r — aH + c), 

 we have 



f(r — a't + c) = — — sin — (r — aH + c), 



A, A, 



TT 27rm . 27T . ., . . m 2ir . . . 



V= — — sm — - (r — a!t + c)+ -oCos — - (r — a't + c). 



and, by supposing that ~— = tan 6, 



2irr 



irm 



v =v 



(l + ^f^(r-a' i + c) + e), 



