Phofessor CHALLIS, ON A MATHEMATICAL THEORY, ETC. 585 



we are unacquainted, such motion should exist, provided it be small compared with the velocity a, 

 we may abstract from it in considering vibratory motion. This will appear as follows. Since 

 equation (5) is linear, we may suppose s to be composed of parts due to separate causes, among 

 which may be included the cause that produces the permanent part of the motion. But the 

 condensation due to this cause being represented by a, we shall have, 



c 



= 0. 



That is, a is either constant throughout the fluid, or is a quantity of an order already neglected. 



ds ds ds 

 Hence the values of — , — — , - in equations (6), (7), (8), remain the same whatever be 

 die dy dz 



the permanent motion. Hence also u — c, v — c, w - c", or the parts of the motion which 



are not permanent, are the same whatever be c, c, c". We may, therefore, either put c = 0, 



c = 0, c" = ; or, suppose u, v, w, to stand respectively for u — c, v — c, w — c" . 



This being premised, let >// = - a'fsdt. Then 



d\L> d\]/ d\lf 



do! dy dz 



and ud.v + vdy + wdz = (d\//), an exact differential. Also by means of equation (4), we derive, 



rf-x// ■ ^ Id'^ d^// d^^ 



-J7 -'^ -b-T + -T^ +^ =0 9)- 



dt- \dx dy- dz' I 



dy 



2. The motions of the aether which correspond to the phaenomena of Light are vibratory. 

 Hence in treating the Undulatory Theory of Light hydiodynaraically, the quantity udx + vdy 

 + wdz must be an exact differential, by what is shewn above, without referencL' to the manner 

 in which the fluid was put in motion, the reasoning being prior to, and entirely independent of, 

 any such considerations. The condition of integrability is to be satisfied generally. One obvious 

 method of doing this, is to suppose the motion to consist of separate motions which tend to 

 or from fixed centres, and are functions of the distances from the centres. But the pha;nomena of 

 I^ight do not accord with this supposition, since, instead of spreading equally in all directions 

 from a centre, it is generally propagated in the form of rays. Another way of satisfying the 

 condition of integrability in a general manner, is to suppose \|/ to be the product of two functions 

 <p and /, such that (p docs not contain x or y, and / does not contain z. For on these 

 suppositions, 



df df _ _ ^ dtp 



dy' 



idf df \ dd) , 



udx + vdy + wdz - (h l-^- dx + , dy\-^f--dz, 

 ^ \dw dy I dz 



which is an exact differential of fcf) with respect to co-ordinates. The consequences of thus 

 satisfying the condition of integrability, which are of a very remarkable kind, I now procied 

 to develope. 



.3. The above values of n, n, w give, 



du d'f ^ _ , <f dw d"(j) 



d^ " *^ dx^ ' d^ ~ "P dy-' dU ~ ^ dz' ' 

 Let us now, for a reason that will presently appear, suppose tliut / does not contain/. Then 



ti = (b-j-, v = (p~,w=f -^, and 

 dx ' du da 



