174 Mb CHALLIS's RESEARCHES IN THE THEORY 



SECTION I. 



On the Integral of the Equation -j-^ + -~ = Q. 



2. This equation is applicable to all problems respecting the motion 

 of incompressible fluids, which require for their solutions the consideration 

 of motion in one plane only. Mathematicians have obtained integrals 

 of it suited to the particular questions they were discussing ; for instance, 

 in solving the problem of waves propagated in a canal of uniform 

 width, M. Poisson has given a value of (p, which, while it satisfies the 

 equation in question, is exclusively applicable to that problem. But 

 it is well known that by the common method of finding the integrals 

 of linear partial differential equations of the second order between 

 three variables, a value of cp may be found prior to any consideration 

 of the circumstances under which the fluid was put in motion. There- 

 fore any inferences respecting the nature of the motion, which may be 

 drawn from this integral, must be equally applicable to all problems of 

 this class. To obtain such inferences is the object of the following 

 reasoning. 



3. The integral I speak of is. 



To ascertain its general signification, I propose to determine the forms 

 of the functions F and jf, independently of any hypothesis respecting 

 the mode in which the fluid was put in motion. The quantity (j) is 

 subject to the condition {d(p) = udx-^vdy, where u and v are the 

 velocities at the poiiit xy in the directions of the axes of x and y 



respectively. Hence ^^=«, -j^=v, and 



u = F' {x + y^/'^\)+f {x-y^/'^), 

 v=V^lF'ix + y\/^^)-\/'^fix-yV^l). 



