818 Mr. J. L. Woodbridge on Turbines. 



The sum of the quantities in the second column will be 

 zero ; hence 



mco 2 psmy—m — '-j xp sin 7 ~ dpd0 = O. . (1) 



Substituting 



— = v sin y, and xpdOdp = -k-, 



and dividing by m sin 7, we have 

 Integrating, 



ay 2 pdp—^dp = vdv (2) 



[^"1 limit r ~| limit 

 *«v-fj =M ■ • • • (3) 

 OJ limit L. J limit 



The sum of the quantities in the first column gives the 

 pressure normal to the vane, which multiplied by p sin 7 

 gives the moment. This done, and substituting as before, we 

 have 



eT 2 M = mv cop((~ a cosy — 2 J— pv sin 7-^ + ^0087 — p — J* -~ sin 7. 

 Putting mv sin 7= — dddp, where Q is the quantity of water 



LIT 



flowing through the wheel per second, and integrating in 

 reference to 6 between and 2tt, we have 



dM==SQ[©^fi)cos7-2J-/wsin7^ + t;cosy-p5^^J^. 



Multiplying (2) by - cos 7, we have 



co 2 p 2 7 p cosy dp 7 dv , 



which, substituted above, gives 



<2M = SQ — 2wpdp + p cos y-r- dp + vcosydp — pv sin y -^ dp , (4) 



and integrating, 



M = SQ[— (op 2 + pv cosy] 

 = -SQp[cop-vcos 7 ]?Z .... (5) 



But o)p — v cos 7 is the circumferential velocity in space of 



