APPLICABLE TO THE MOTION OF FLUIDS. 395 



which are x = I, y = k, z = 0, we have I = n. Hence mh = —I, and 

 h = . The equations of pq r thus become x = m% + l, y = p{%+ — \. 



Now the co-ordinate Os of the point p is / + r + Sr. Hence, the first 



Sr 

 equation of Pp gives I + r + Sr = a% + I + r, or % = — , and conse- 



tit 



quently from the second, y — -.Sr. These are the other two co- 

 ordinates of the point p. By substituting these values of the co-ordinates 

 in the equations of the line pqr, which passes through p, we shall readily 



find that m = a + -$- , and p — =-i J- . 



Sr r I + r + Sr 



If now x = a'z, y = b'% be the equations of a line drawn through 



a' 

 O parallel to AZ, the cosine of the z NPO = ... ; and the 



V 1 + a' 2 + o" 



cosine of the i npr — — == — . " = . Hence by substi- 



p n/1 +«" + ** s/l + m T +p* J 



tuting the values of m and p, expanding to the first power of Sr, 



and remembering that 1 + aa' + b b' = 0, it will be found that, 



a' / I + r + bb'r . \ 



cos / apr = —7============ . 1 + j—tj r- Sr . 



v/1 -i- a' 2 + b' 2 V aa'r{l + r) ) 



Let V and V + dV be the velocities at P and p at the same 

 instant, and let w and w + Sw be their components in the direction 

 of the axis of as. Then 



Vd 



w = P'cos z NPO = 



v/l + a' 2 + *' 2 ' 

 (F + SV)a ( I + r + bb'r 



and w + 3?p = (r+ SF)cos * npr = \ i , ' .,. 1 + 7-77- — r.^r 



7 r ^/\ + a ' 2 + b' 2 \ aa r(l + r) J 



V(l+r + bb'r)Sr dSV ... . , 



Consequently, Sw = — , 7 v . — . = + . , == ultimately. 



H " ar(l + r) v/l + a' 2 + b n \/l + a' 2 + b'* 



