34 Proceedings of Royal Society of Edinburgh. [dec. 20 , 
velocity - Y in direction Ox to be impressed upon the solid and 
fluid, so as to bring the former to rest, and let <£ be the velocity 
potential of the fluid motion relative to the solid at any point P 
whose polar coordinates are r, 0. Then <£ satisfies Laplace’s 
equation a 2 <£ = which changed to polar coordinates becomes 
1 P 4 , 2 <20 1 d f . \ _ „ 
dr* rdr + r* sin 6 d6\ d& ) ~ 
We may take for a solution of this equation 
<£= - Vr cos <9 + a 0 — + a 2 P 2 -§ + a 4 P 4 + &c., 
where P 2 , P 4 , & c., are Legendre’s coefficients of even order, and a 0 , a 2 , 
<fcc., are constants to be determined. The first term is introduced 
because the velocity at infinity parallel to OA must be - Y. We 
can determine a 0 , a 2 , &c., by the condition that d<j)/dr must vanish, 
when r = a. 
This gives us 
V cos 6 = - - (a Q + 3« 2 P 2 + 5a 4 P 4 + &c. ) . (4) 
CL 
This must be true from 0 = 0 to 0 — ^ r. 
ISTow cos 6 can be expanded in a series of Legendre’s coefficients 
of even orders between the above limits. Putting cos 0 = x, 
^ rrP m,dx — q. \ ^ nPndx — 0 , 
which last is true if both m and n are even. We then get 
and generally 
a 0 = “ 2 * a 2 = “ » 
r + 1 2.4. . .(r + 2) 
It will be found that the differential equation to the stream 
a 
lines can be expressed in the form (putting — = p for shortness) , 
(1 x?) | Y^ 3 a 2 p d * + a 4 p 3 + &c. | dp 
| ^2 Vo£c + a 0 + 3a 2 P 2 p 2 + 5<x 4 P 4 p 4 + &c. j<i«£ = 0. . (5) 
