531 
of Edinburgh, Session 1877-78. 
where a Y , a 2 , a 3 are scalars , which separately vanish when the 
systems are isothermal. 
Expanding the last equation we have 
— 1 qig~ l -f Vq . q ~ l . qiq~ l + qiq~ l Vq . q~ l - 2S (. qiq~ l V ) q . q~ J = - 1 
u i 
or, writing 
qiq-i = % , 
i' + 2 Si'V? ?-> - 2 S(i'V) q.q-^'h. 
We obtain Dupin’s Theorem in its most general form by operat- 
ing by Si', S .j' f S.&' on this and the two similar equations respec- 
tively. It is thus expressed as three equations, of which one is 
Si'S(rv)2.? _1 = 0. 
Again, by multiplication by i ' , and by adding the other two 
equations multiplied by / and k’ respectively, we obtain also 
2 — + 22«'S*'— 1 - 2 Y.Vqq- 1 + 2 Vq . q~ l = 2 
n u x 
or 
2^ + 2 V.Vqq-' + 2V ? .o->= -2^li 
M U Y 
whence 
S.V? q-' = 0 , 
and 
S — + 4 V? . q- 1 = - . 
u U 1 
When the systems are isothermal as well as orthogonal, this 
equation may be put in the singular form — 
The results given in this section were laid before the Society in 
May 1876, but were mislaid, with other papers then read. 
1 (8.) The great desideratum in the application of quaternions to 
problems such as those just treated, seems to lie in the discovery of 
the general solution of the equation 
Vr = 0, 
I 
where r is a quaternion. Unfortunately this seems to depend 
ultimately upon Laplace’s equation, treat it how we may. It is 
