218 Mr EARNSHAW, ON FLUID MOTION. 



completel}' distinguish the fluid motion now under consideration from 

 all that has yet been investigated. 



Eaeli set of values of f, g, h will furnish eight independent arbitrary 

 functions in the expression for (p, but for reasons precisely similar 

 to those advanced in Art. 8, we need only consider one set at a 

 time, which we know denotes a possible motion; the possibility of the 

 coexistence of- any number of such sets in the value of <p being 

 physically interpreted by saying, — that the fluid can simultaneously 

 transmit anj' number of disturbances, each as though the others did 

 not exist. 



In Art. 13 it was mentioned that one arbitrary function alone 

 could not represent fluid motion ; but in the case before us, each 

 function represents a possible motion, and it will be shewn hereafter 

 that the eight arbitrary functions furnished by one set of values of 

 /,' g; h denote as many independent waves moving in different direc- 

 tions: and therefore no generality will be lost by employing only 

 one of the eight functions; and the reasoning upon it will be ap- 

 plicable to the other seven; and the full effect denoted by one set 

 of values of f, g, h will be seen by supposing the eight waves cor- 

 responding to the eight functions to coexist and to be transmitted 

 simultaneously in the medium. 



19. We are at liberty to take any one of the eight functions as 

 a general representative of all, suppose then that 



cp = F \f{x-a) + g {y- li) + h{%-y)-at, f, g, h} (1) 



The origin of co-ordinates is a fixed point in space ; but let it be 

 transferred to the point X, Y, Z, which will be done by writing 

 X + x', Y +y', Z + z for x, y, z respectively, and then 



cp = F {fx ^ gy' + M +f{X-a) ^ g{Y - ^) + li{Z--i) - at, f, g^ h\. 



Now as the new origin is arbitrary, we will suppose it moveable 

 instead of fixed, and that it moves in such a manner as to satisfy 

 the equation 



/(X- a) +g-(r- /3) + /^(Z- 7) = «/ (2) 



which supposition is allowable because f, g, h are possible quantities. 



