1906-7.] Note on Quaternion Integral Theorems. 
371 
XXXVII. — Note on Quaternion Integral Theorems. By Heinrich 
Hermann, D.Sc. (Tubingen). Communicated by Professor C. G. 
Knott. 
(Read July 15, 1907. MS. received September 21, 1907.) 
The transformations of volume integrals into surface integrals, and of 
surface integrals into line integrals, are usually established first for a small 
parallelepiped and parallelogram.* These elements are not, however, very 
convenient, or indeed altogether satisfactory, for the extension of the 
theorems over finite domains. If arranged in straight rows and plates, 
the elements cannot be fitted to a general boundary ; and if they are 
arranged in curved rows and shells like the stones of a vault, there will be, 
in general, between various elements of the first order, spaces of the second 
not represented. This inconvenience disappears if tetrahedra and triangles 
are taken as elements. In Quaternions, the theorems are easily proved 
for such elements. 
Let a /3 y be three small vectors drawn from the point 0 and forming a 
tetrahedron, the vectors being arranged in the positive order of circuition. 
Let q be the value (generally quaternionic) at 0 of a function of space. 
There exists the identity — 
Sa/?y . V q = y.Y /3ySa V . q 
a i Pi y 
which may be written — 
Now — -g-S af3y is the volume of the tetrahedron; the first term on the 
right side is the vector area of the triangle a, f3, y multiplied by the value 
of q at the centre of gravity of the area ; and the second term contains the 
corresponding quantities on the other faces of the tetrahedron. Hence we 
may write the relation in the form — 
dvV q = 2 (dvq ) 
where dv is the volume and dv the vector area measured outwards. 
* See M‘Aulay, Utility of Quaternions in Physics , p. 19 ; and Joly, Manual of Quaternions , 
pp. 71, 215. 
