'208 Mr EARNSHAW, ON FLUID MOTION. 



F and F^ enable us arbitrarily to fix the epoch from which the time 

 is reckoned, and further to accommodate the wave-surfaces to any pro- 

 posed form. 



These observations will be fully illustrated in a subsequent part of 

 this paper. 



8. The object to which it will be necessary first to turn our attention 

 in the above integral is the discovery of the meaning of the constants 

 .f, g- Whatever forms be given to F, F„ whatever origin be taken 

 for co-ordinates, whatever epoch for the time, still J" and g are un- 

 affected: and as an infinite number of quantities fulfilling the condition 

 ,/"" + ^" = may be invented, and any one set will satisfy the equation 

 d/(p + dy(p = 0, which in a general view of the question is the only 

 further condition to which they can be subjected, it follows that all 

 imaginable values of J" and g ought equally to appear in the general 

 integral (see Art. 27) ; one set of values giving only a partial solution of 

 the proposed differential equation. Hence the general integral of the 

 equation of continuity of a moving fluid of two dimensions is 



<p = F,{f,{x- a,)+g,{y- (i,),f,, gu t} + F; {f,{x-a,)-g,{y - (i,),f„ g„ t} 

 + F, {fi {x - a,) +g.. {y - A), /„ g„ t} + F,! {/. {x - a.) -g, {y - /3.), ^, g,, i} 

 + &IC 



the quantities of fi', //, //.... ^V, gi, gi.... embracing all values from 

 - 00 to + 00 . It is manifest, however, that inasmuch as each set of 

 values can be separately made to satisfy the equation of continuity, 

 each set will represent a possible motion, /. e. a motion of such a nature 

 that the fluid can transmit it. Hence the general integral just exhibited 

 fiunishes us with the following physical fact, which I believe has never* 

 before been fully accounted for; — 



* It has been remarked that (p may be represented by F, (.r +y J - 1) + F^ (.r + y J - '[) + .. . 



+/. (.■^ -y 7^) + fo (•'■ -y J^) + • • • 



and thence the superposition of disturbances has been inferred: but before this principle can 

 be inferred, is it not necessary to shew that F^ (x+y J - l) + F,{x +y J — 1)+ . . . 



