38 Proceedings of the Royal Irish Academy. 



As regards formula 661, what is contemplated is a variation of the rate 

 of turning of the tangent at and about | , and if the mean of this variation 

 over the range U, be c \ : i . then 8x' = 8(<*x' ^ &■ Hence ( 66 ) is 

 equivalent to 



N .v. in spite of the greatness of log e?, the product c£ log e? -> as 

 c£ -> 0. and thus the formula shows that cq tends to vanishing smallness in 

 comparison with c \ ; . A corresponding result holds for <?;<?. 



Thus :: :hat infinitesimal variations of the rate of turning of the 



tangent, of the kind contemplated, corresponding namely to alterations of 

 curvature which are infinitesimal and do not introduce corners, may be 

 regarded as leaving the velocity q unaffected. And therefore a variation 

 which in - j? increases the curvature, and conversely. 



is according! ■ . .ite now to express in geometrical form the 

 principle obtained in the preceding article, namely in the statement that 

 the more the curvature of the sides of the obstacle is brought into the 

 neighbourhood of the prow the less will be the divergence of the wake. 



Of :nplete concentration of curvature at the prow must be ruled 



out aid reduce the obstacle to a plane lamina edge-on to the 



- than resistance to the flow have to be taken 

 account of, and i i the degree of concentration of curvature near the 



prow . the obstacle as much breadth as it requires 



for . >riy other purpose. But the present principle, if it is 



sou: I of bringing the resistance question simply into the 



balance w 1 relevant considerations. 



le principle may be established for an asymmetrical obstacle 

 by considering the formulae got by equating to zero the coefficients of ji in 

 formula- 54 55 5 tjj is negative for curves convex t<"> the region 



of flow. illusion . And the two integral signs 



may be replaced by a single one covering the whole range, provided it be 

 agreed that the sudden change in the value of \ at f = c is excluded from 

 the integration. The equations determining the most forward points of 

 departure may then be written 



£ = -« 



