340 TRANS. ST. LOUIS ACAD. SCIENCE. 



normal pressure, and not the normal pressure, as was stated in 

 the Annals of Math., vol. i, No. 3, p. 44. 



Duchemin applied this formula to the results of experiments 

 made by Vince, by Hutton, and by Thibault, and found it to agree 

 with the results of all better than did the formula, given on page 

 337 that was deduced by Hutton from his experiments alone. 

 (See Annals of Math., vol. i, No. 2.) 



The normal pressure is A^= . — .and the vertical component 



of the normal pressure \%V :=. R cot i. 



-r, ^^ c T7.T R '2' P sin t 



l:'rom the formula JV =1 — — z= ,- .0 • 

 sm I I -\- sni- I 



it is evident that Nz=. P when 2 sin / = i -(- sin^ ?', whence 



sin z =: I, or i — 90°. For all values of i less than 90° N is- 



less than P^ as it would seem should be the case. 



_, ,;- 7-1 • P cot i . , , -.^ . , 



rrom K 1= 7t cot i = ; ^r-^ it is seen that k is equal to 



I -\- cot^ z 



P when cot^ I — 2 cot i ^ — 2. Whence cot i =z i ^ \/ — i. 



A result that indicates that V is never equal to P. V is- 



cot i , • , • , 



equal to zero when — j 5— . :=o, which is true when cot ^ 1= o, 



^ I -|- cot^ t 



or / = 90°, and also when cot i =: cc , or z = 0°. 



-r^ . ^, . . , . P cot I , . , 



Dinerentiatinsf the expression V=z — ; ^=—7 and puttine the 



^ ^ I -|- cot2 /, * *= 



differential co-efficient equal to zero and reducing, we have 

 cot^ / — 2 

 (2+cot2 2)^ .— °* 



Whence we see that Kis a maximum when cot /:= \/2, or z i& 

 about 35° 15'. 



The results obtained by using the formula iV=: — , — ^—^. 

 •^ * I 4- sin2 t 



agree not only with good theory, but also with the results of ex 



periments, and, therefore, it is the formula that should be used. 



in calculating the wind pressure on roofs and bridges. 



