Mr. J. L. Woodbridge on Turbines. 323 



This expression differs entirely from Rankine's equation 

 (10), page 196. But running this wheel at the same speed 

 as Rankine does his, Art. 175, that is, making the final 

 velocity of whirl zero, we shall have 



t' 2 cos7 2 = a>/> 2 , 



and equation (26) becomes 



G)p 2 = cos y 2 \ /r G) 2 p2 2 — 2G)' 2 p 1 2 + 2#H ; 



from which we find 



2+ P -\ tan 2 72 

 Pi 



which is the same as Rankine's equation (3), Art. 175. Sub- 

 stituting these in (27), we have 



2p 2 2 + / ^ 2 2 tan 2 7 2 , 



which is Rankine's equation (4), Art. 175. 



We thus see that Rankine's equations not only do not fit 

 any wheel except the one he is considering, but they apply to 

 that only at one particular speed. These conclusions agree 

 with those in an article on Turbines, by Professor Wood, in 

 the Journal of the Franklin Institute for June 1884. 



In regard to the speed for maximum efficiency, Rankine, 

 in the ' Steam-En gine,' Art. 173, says, " In order that the 

 water may work to the best advantage, it should leave the 

 wheel without whirling motion, for which purpose the velocity 

 of whirl relative to the wheel should be equal and contrary to 

 that of the second circumference of the wheel." Plausible 

 as this appears, it is true only for special cases even for his 

 wheel. Also Weisbach makes the erroneous statement that 

 the velocity of the second rim of the wheel should equal the 

 relative velocity of discharge. Thus in i The Mechanics of 

 Engineering and of Machinery,' vol. ii. page 400 (Wiley and 

 Sons) he says (substituting my notation for his) 



w = 4/co 2 p 2 2 + v 2 2 — 2v 2 cop 2 cos 7 2 = /y/ (cop 2 — vj 2 + 4cop 2 v 2 sin 2 ^ 



