372 Proceedings of the Royal Society of Edinburgh. [Sess.i 
By the usual extension this gives — 
/ / 'fdvVq- j jdq 
Similarly, the identity — 
V( Va/3) V . 2 = ( aS/3 V - £Sa V )g 
may be written — 
yI“/ 3 V.« 2 = a( 2 -S^V.g) + (/3-a)( ? -8^V.2)- i 3( 2 -s|.V.g) 
Each term on the right represents one of the vector sides multiplied by 
the value of q at the centre of the corresponding side. The relation may 
therefore be written — 
V dv V . q=^dpq 
for a small triangle. 
By the usual extension this gives — 
j JvdvV . q = jdpq 
[Tait’s original proof of the second theorem ( Quat ., 3rd edition, § 498) is 
fundamentally the same as the one here given, but it is probably more 
difficult to follow at a first reading. — C. G. K.] 
(. Issued separately November 8, 1907.) 
