118 
The simplest way of finding Dp is as follows. We have 
seen that 
DjO- = O’ - (So'V)o’ = <7 - SYo-p - 
operate upon this with YA, then 
YvDto- = p - 2YvYo-p = p - (Sffv)p + (Spv)o- 
= D,p -f (Spv)o- 
But from the circulation YvD^o- = 0 
.'.D,p= -(Spv> YIL 
The value in terms of k and \p is not so simple as to deserve 
notice. 
The geometry of the motion is not easily explained 
owing to the fact that </> is not the third surface satisfying 
= Oj also k and \p will not generally be independent, as 
the condition for irrotational motion is that they should 
touch at the points of intersection. 
‘‘Notes on some Quaternion Transformations,'’ by R. F. 
Gwythee, M.A. 
The following theorems are frequently required in physi- 
cal problems, especially in the motion of fluids. 
I. 
If T denote the vector of any point and v Hamilton’s 
operator, and if p and o- are any vector functions of r, then 
V(p(y) = vpo- - pV<7 + 2(Spv)(T I. 
More generally — If p and <t be any vectors depending on the 
scalars a, 6, c, &c., and if a, j3, y, &c., be any vectors what- 
ever; and if + yc^c + &c. = D, where denotes 
differentiation with regard to a, &c.. Then 
D(po’) = Dp.ff — pD(T + 2(SpD)(T. 
For ad^ipcr) = ad^fp.ar + ap.daV 
= adap.cr — pa.dg(T + 2(Spada)<T 
since ap + pa = 2 Spa 
If we now form the similar quantities for &c., and add 
the respective sides of the equation thus formed 
.*. B(p<t) = Dp.a — pD(r + 2 (SpD)(T 
II. 
