1886.] Rev. H. J. Sharpe on Motion of Compound Bodies . 35 
Remembering the known relation 
if 1 -* 2 
) vf} +»(»+!) p «=°. 
and examining the general term of (5), we see that (5) is an exact 
differential. Therefore integrating we shall get for the equation of 
the stream lines 
(1 - x 2 ) 
aV 
> 2 ^P 
a, 
2 P 
2 + a 
4 
2pj 2 2 dx 4 4 
-f &c • l - = C 
For the particular stream line, which passes through the point 
B, the above must he satisfied by p= 1, or £c = 0, and 
aV 
therefore C = — > so that its equation is 
(1 - x 2 ) 
pf_ 
a * 2 dx + * 4 
dP* 
c/.r 
f &c. 
aV 
— cIqX 2 
(6) 
For distant points p is small, therefore for such points neglecting 
all the positive powers of p in the above, we can readily prove that 
y — J(f)a is an asymptote, and that the curve lies below it. It is 
interesting to see whether at B the curve goes up or down. To this 
end, in (6) put a + 8r for r and Jtt + SO for 0, expand and retain not 
beyond the first power of Sr and the second power of SO. It will 
he found that we shall get 
2 Sr - aSO 2 + a 
1 4w + 1 
ft 2 ft . + 1 
~ i + f + + &c. 
3.5. . .(2ft - 3) 3.5. ..(2ft + 1) 
2.4.. .(2ft + 2) 2.4. ..(2ft- 2) 
+ &c. 
S0= 0 
( 7 ) 
In order to get the terms after the 3rd in this series, we must 
take ft from 3 to infinity. There seems to he no doubt that the 
value of the series in the square brackets is zero, although it is 
rather difficult to prove it algebraically, for the expansion (4) is 
true even at the limits. Moreover the expansion (4) is never 
differentiated, therefore def)/ dr is zero for r = a even at the point B, 
therefore, as in the case of the circular cylinder, the stream line 
through B must touch the circle at B ; assuming therefore that the 
series in (7) is zero, the remaining terms show us that the curvature 
at B is upwards, and the radius of curvature equal to a. The same 
figure therefore as was employed for the circular cylinder will do for 
this case, provided that we make OC = J(2)a instead of 2a. 
