194 Proceedings of Royal Society of Edinlurgli. [sess. 
tyro in integration has to take to fix in his memory the values of 
or d tan &c. 
The only peculiarities of the question seem to he due to the 
contrast between the (apparently) great generality of the initial 
equation and the extremely restricted character of the sole solution. 
This will he abundantly evident from the discussions which follow, 
since it would almost appear as if the conditions arrived at were too 
numerous to he simultaneously satisfied. I find it very convenient 
to use a symbol 3 ^in the sense of to express rate of in- 
crease per unit of length. Thus 
may he written 
V = 
,d .d n^d 
5 
y = a0j -1- /502 + /503 , 
where a, y are any rectangular unit-system. 
The equation 
d(T = uq~^^dpq , . . . . . . ( 1 ) 
(where u is a scalar, and q a versor, function of p) ensures that an 
element of space at o- corresponds to a similar element at p ; so that 
the transformation from p to o-, or vice versd^ is from one mode of 
dividing space into infinitesimal cubes to another. [From the 
purely analytical quaternion point of view the question may he 
regarded as simply that of finding u and q as functions of p, so that 
the right-hand member may he a complete differential.] We have 
at once 
Sac^o" = - Sc/p y . Sao- = u^.dpqaq~^ 
whatever constant unit vector a may be. Thus 
- y Sao- = ..... (2) 
A part, only, of the information given by this is contained in 
YV.w^a^-i = 0, (3) 
