226 Proceedings of Royal Society of Edinlurgli. [sess. 
is simply the quaternion identity 
47rV-2(o = 47rV-iV-ico 
= 47t V -^VV-i(o + 47 t V -'S V -'(0. 
Similarly the equation 
47t Pot u = - Max New ih 
is a travesty of 
4:rV-% = 47rV-iy \i) 
and the transformation 
O) = VV“^w 
= VVV-iw + VSV- (o 
\ 
is hidden from sight in the symbolic mask 
47ro> = V X Lap o) - VMax w. 
If these and other like equations are to be of any service, what 
brain is to be expected to carry the memory of them in their Max- 
Lap-New-Pot forms, which vary according as w is solenoidal or 
irrotational ? To the ivorker with the quaternion V, on the other 
hand, there is absolutely no difficulty. The functions will come in 
as they are needed, and transform into useful shapes by the simplest 
laws of their being. And all this exquisite potentiality of trans- 
formation Gibbs wilfully throws to the wind, because a quaternion 
is not a vector. 
There seems little doubt that, in leaving the Hamilton causeway. 
Professor Gibbs has here fallen into a veritable slough, through 
which he has himself presumably passed laboriously enough, but 
wherein the hapless student who would follow will quickly reach 
the depths of despondency. No finer argument in favour of the real 
quaternion vector analysis can be found than in the tangle and the 
jangle of Sections 91 to 104 in the “The Elements of Vector 
Analysis.” 
It is little short of marvellous that, with such an appropriate 
and expressive symbolism and calmlus as Hamilton has put to our 
hand, certain writers should think that anything is gained either in 
lucidity or completeness by notations like “ Curl Pot o>,” “ Div Pot o>.” 
To the uninitiated Curl and Div are as unintelligible as VV and SV; 
while to the initiated the latter forms are infinitely to be preferred, 
simply because they belong to a complete and consistent system of 
