36 Proceedings of the Royal Irish Academy. 



is simply this : — What kind of relation between \ and £ will make the negative 

 range of integration with respect to \, that is from \ = pit to \ = \(a), as 

 great as possible numerically, so that x(«) may be as small as possible ? 



When the curved sides of the obstacle are convex to the stream d% is- 

 negative throughout the range from £ = to £ = a. And as ay(a - £)- is 

 greater than unity, the whole range must be numerically less than pir, so 

 that \(") i s necessarily positive. Thus the stream-lines must diverge; but 

 the divergence may be kept small. 



If formula (Gl) be written 



P* = Limv;,,' (a - |)i] (- S X ), (62) 



it is clear that if the greatest values of -B\ be associated with the smallest 

 values of («V( rt- ?J^I they contribute less to the sum, and therefore a 

 greater range of - S,\ * s required to bring the sum up to an assigned value. 

 So the association of the greater values of -S\ with the smaller values of £ 

 diminishes the divergence of the wake. In other words, if the rate of turning 

 of the tangent to the curved side of the obstacle l>e greatest near the prow, 

 this configuration makes for reduction of the divergence of the wake, 



22. It will be noticed that in the last sentence, where the word 

 "curvature suggests itself, a different phrase has been employed. "1 1 1 is is 

 because the argument Ims been founded on a functional relation between \ and 

 5, and the "rate of turning of the tangent " which appears in the result is not 

 -,/ v ./.. where - is the arc of the contour, but -d\/d^. If a Lb the curvature 



of the obstacle, 



a = -dxJds = -<./ v <t>y, 



where y is the resultant velocity; and the theorem cannot 1" i xpressed in a 

 purely geometrical form until it d how change in the value of 



->\ •/{; affects the value of q. It cannot be assumed a priori that -dy}d% 

 and i increase or decrease together. 



It will be advantageous to write d\ for -d\, remembering that this is 

 always positive on the convex obstacle. 



If an addition B\ be made to <l x ', and if this be < centrated at a 



single point £ = £ of the contour of the obstacle, a corner is created there, 

 and q becomes either zero or infinite. It is therefore necessary to 

 suppose S\ to be spread over a range of £ Buri-ounding the point £„, say 

 from 5,-7 to £, + 7; if 2y = 8£, it will he assumed that r X ' and c£ are 

 infinitesimals. If £ = £„ + A, the addition to <<\ over the postulated range 

 may be represented by rl 0(A), where 8 is a function such that 



« ■ \: 



dO(\) = 0(y)-O(-y). 



