Mr EARNSHAW, ON FLUID MOTION. 209 



Any number of disturhances separately, though simultaneously, excited 

 in a fluid medium, will he separately, independently and simultaneously 

 transmitted through the fluid, each as perfectly as though the others did 

 not exist. See Art. 4. 



9. Having ascertained this to be the meaning of the integral in 

 its general form, it will be sufficient now to consider the transmission 

 of one disturbance only; and if this investigation be carried on upon 

 the general hypothesis of a single disturbance of any kind affecting the 

 fluid, the results Avill be of a general character also. This point'' will 

 be gained by keeping our integral under the form 



= ^ {/{oc-a) +g(y-(i),fg^ t] 

 + F,{f{x-a)-g{y-(i),f,g, t\. 



From this we shall proceed to deduce the following results. 



I. Motion cannot be represented by otie of these functions alone. 

 For, if possible, let motion be represented by 



^ = F\f{x~a) +g(y-(i),f^, f^ 



or, for brevity, by cp = F. 



Then the (velocity)^ of the particle whose co-ordinates are (x, y) would 

 be 



= {d,(i>Y + (d^<py 



=f\F'' + g\F" 



that is, the medium is at rest. F' is used to denote the differential co- 

 efficient of F with regard to the quantity /(a; -a) +g(y-fi). 



is a more general expression than one function F{x+yJ^i)} I„ the Jnlegral Caiculus, we 



know that C^ + C+C^... represents only one constant C: are we certain that F, + F^+... 



represents more than F> are we sure that F„ F„ F,... are so essentially distinct that they 

 cannot be united in one fijnction? 



