the Resistance of the Air to an Oscillating Sphere. 231 

 Hence 



This determination of the arbitrary factor N is the analy- 

 tical proof that the assumed kind of motion is possible. 



Let us suppose, for example, that the surface of displace- 

 ment is a spherical surface, whose equation is 



{x - a)* +{y- bf + (z - cf = R 2 , 

 R being its radius, and «, b, c the coordinates of its centre. 

 And first, let a, b, c be constant, and R be a function of the 

 time. Then 



d t at 



2 R 

 Hence N = -^-, and consequently N is a function of the 



time. 



■ Next let a, b and R be constant, and c variable. This 

 is to suppose that the surface of displacement is a spherical 

 surface of given radius, the centre of which is moving parallel 

 to the axis of z. For this case we have 



df „, .dc , T , z — c dc 



^=-2(,-c)^;andV = -j r .^. 



2R 2 



Hence N = 



t \ dc ' 

 <* " c > Tt 



The supposed surface of displacement being thus shown to 

 be possible, it necessarily follows, that if a smooth solid sphere 

 move rectilinearly in the fluid, its surface coincides with a 

 surface of displacement. Here then we have the case of the 

 oscillating sphere, and as z — c occurs in the expression for 

 N, we have come to the important conclusion that udx 

 + v dy + w d is is not an exact differential in this instance, 

 when the variation of the coordinates is from one point to an- 

 other of the surface of the sphere, and therefore not an exact 

 differential for every variation of the coordinates. It becomes 



so when multiplied by , if R be supposed to vary with 



2 — c 



X, .?/, z. 



Let us, therefore, suppose in general that 



N (u d x -J- v dy 4- to d z) = d <p. 

 Then 



_ 1 d$ 1 dtp 1 dtp 



"-¥-Tx"> VSS K-dy"> W ~'W i d~z' 



