N OF CKUTAIN FDK.MS OF SII.M TIM-. 



diameter. This \\ill undoubtedly be the case, ami M. Bo ussi NESS'S proof will hold, if 

 the boundary "I" the section be everywhere convex, for then it is evident that each 

 stream-line will consist of a single simple oval (fig. 10). But when we deal with 

 sections like those of the present paper it is by no means so clear that this will be 

 the case. There may be stream-lines which consist of two separate ovals (see fig. 11), 

 and it then Incomes very much open to question towards which part of the section 

 the lines will be most crowded. In fact in some cases, as we have seen, this' will 

 occur at intermediate points, F (fig. 11), and in other cases the narrowness of the 

 nick or the sharpm^* of the bend will counterbalance the limited number of the 

 stream-lines through the neck, and the fail-points will be at B, B. 



19. Comparison with SAINT- VENANT'S Results. "Efficiency" of the various 



Sections. 



When we compare the results given by these sections bounded by confocal ellipses 

 and hyperbolas with those given by DE SAINT- VENANT in his edition of the ' Leons 

 de NAVIER' for the "sections en double spatule," we find very good agreement. Thus 

 the critical value of the ratio of breadth of neck to length for which the fail-points 

 split up each into two is given by SAINT- VEXANT as '3247. For the sections of this 

 piper, when $ = 7T/6, I find this critical value to be '3215. 



The following are the principal numerical points (see ' Lecons de NAVIEH,' p. 365) : 



SAINT-VEXANT'S section, 

 Section = T/2, ft = >r/6. , ft = . 2Q 



Ratio of breadth of neck to length .... 



M .MO 



S/& (S lieing maximum stress) 



E = ; (M/M,)/(S/8o) 



Ifatio of distance of fail-points from short axis 

 to length of longer semi-axis 



2173 

 4443 

 9144 

 4859 



3449 



20 

 3921 

 8668 

 4524 



4561 



On the whole the sections of the present paper appear the more useful, M/M and 

 the quantity E being greater than for in: SAINT-VENAXT'S mil sections. This 



