174 MR. R. F. GWYTHER ON SOME 



Operate upon tliis with Vy^ then 



VvDfO-=p— 2VvVo-p=p— (So-v)p+ (Spv)o- 

 =D,/)+(Spv)o-. 

 Butj from the circulation, V vDiO- = o ; 



.'.J)tp=-{^pV)o- (VII.) 



The value in terms of k and '\fr is not so simple as to 

 deserve notice. 



The geometry of the motion is not easily explained, 

 owing to the fact that (j> is not the third surface satisfying 

 Di(a?) = o; also k and i/r will not generally be independent, 

 as the condition for irrotational motion is that they should 

 touch at the points of intersection. 



XX. Notes on some Quaternion Transformations. 

 By R. F. GwYTHER, M.A. 



Read February 24tb, 1880. 



The following theorems are frequently required in phy- 

 sical 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 a are any vector functions of t, 

 then 



V(po-) = V/3o--pVo- + 2(SjoV)o- (I.) 



More generally, if p and a be any vectors depending on 

 the scalars a, b, c, &c., and if «, ./3, 7, &c. be any vectors 



