Manchester Memoirs, Vol. xlii. (1898), No. 8. 



VIII. On a General Method of determining the Form 

 of the Velocity-Potential of Fluid-Motion in Two 

 Dimensions across a Channel with Straight Sides. 



By R. F. GWYTHER, M.A. 



Received and read January nth, iSg8. 



In a paper'"' on the permanence of Mathematical 

 Forms of the expressions for physical quantities, I have 

 pointed out that the forms of the general equations in 

 applied mathematics are necessarily limited by purely 

 mathematical considerations, and, to that extent, indepen- 

 dent of the physical assumptions on which they may have 

 been originally based. 



The present paper may be regarded as a continuation 

 of that paper, but dealing with the limitations imposed by 

 the same considerations upon the possible forms of the 

 solutions of the general equations rather than upon the 

 forms of the equations themselves. 



It will appear in the development of the subject, that 

 when the form of solution is limited by the considerations 

 proper to a problem, the process of change of coordinate 

 axes, or of resolution of velocities, &c., will be notably 

 simplified in consequence. 



It is, of course, necessary to select an illustration only 

 of the method proposed, partly because the conditions in 

 different problems are so varied, and partly in order to 

 make the character of the method clear by making the 

 conditions definite. 



For these reasons, I have selected, as a general 



* Manchester Memoirs, Vol. ix., 4th series, 1895, P- IIQ* 



■^JmL 13th, i8g8. 



