344 
Proceedings of Royal Society of Edinburgh, [may 21, 
C' - B" = {a\ c"){c, d, c") - (a", b'\ c"){h, b\ V) 
= a'c' + &V + dc" - a"b - h'h" - b"d’ 
and thence, writing 
p = a{d - b") + b{a” - c) + c(b - a ') , 
= ad — ab" + a'b - a'c , 
we have 
C' - B" + p = (a + &' + d'){d - h") = 0 
if a + b' + d' = O', viz., this being so, C' - B" = - p, or the first 
equation is D{d -b") = p, = a{d -b")-]rh{a” - c) + c{b- a'), that 
is = +>?—+/— , the first equation of Helmholtz, and we 
dx dy dz 
thus have the equations of Helmholtz, if a - 6' + c" = 0, that is if 
du dv dw 
dx dy dz 
The foregoing three equations J){d - b”) + C' - B" = 0 , &c., are 
the quaternion equation {(r = iu+jv + hw , y 
^ = D , denotes a complete differentiation). 
_Wcr = VVi<r2S(7iV.2 
of Mr M^Aulay’s paper “Some General Theorems in Quaternion 
Integration,” Messenger of Mathematics, vol. xiv. (1884) pp. 26-37; 
see p. 34. 
3. The History of Volcanic Action during the Tertiary 
Period in the British Islands. By Archibald 
Geikie, F.R.S. 
{Abstract.) 
In an introductory section of the paper, a sketch is given of the 
literature of the subject, and reference is made to the labours of 
Jameson, Macculloch, Berger, Boue, Oyenhausen, Von Dechen, 
Hecker, Zirkel, Judd, and other observers. The author then men- 
tions the progress of his own investigations, which were begun 
some thirty years ago, and of which the first published part ap- 
peared in the Transactions of the Society for 1861. A journey 
into Western America in 1879 gave him new insight into the 
problems presented by the youngest volcanic rocks of Britain. 
