( 675 ) 
XXV.—On Minding’s Theorem. By Professor Tarr. 
(Revised June 23, 1880.) 
The following paper contains a short digest of investigations communicated 
to the Society on several occasions during the past, and the present, session. 
The work had been for some months laid aside, but my attention was recalled 
to it by. Professor CurysTAu’s valuable paper, in which he treats MINnDING’s 
Theorem as an example of PLicker’s methods, and also by the help of Rop- 
RIGUES co-ordinates. Iam induced to publish a few of my results in full, as I 
think that a comparison of the analysis employed by CurystaL, with the very 
different analysis employed by myself, may be useful as well as interesting, 
especially from the point of view of the simplicity of the quaternion method. 
Even when the quaternion processes are written out at full length, they are in 
general shorter than the most condensed forms of ordinary analysis ; and there 
can be no doubt that they are much more easily interpretable into the corre- 
sponding geometrical ideas. 
A hastily-written proof of the main theorem, somewhat on the same lines 
as the first of those now given, was printed in the “ Proceedings of the London 
Mathematical Society,” No. 147. But the present version is much simpler ; and 
it is requisite for the intelligibility of the rest of the paper which, I repeat, is 
given mainly for the sake of the quaternion processes involved. 
I commence with a few preliminary transformations. This would be alto- 
gether needless if quaternion methods were at all as familiar to the majority of 
mathematical readers as are the more usual ones. 
1. In what follows we have a good deal of use to make of certain properties 
of linear and vector functions, so that some of the less obvious of them are here 
briefly stated. 
Let a,, ao, &c., B,, Bo, &c., be any two sets of vectors, and let us consider 
the vector 
K=2V Ba. ; : : ‘ : (1) 
If we operate by V.c, where o is any vector whatever, we have 
Vor=V.crVBa 
= 2(aSBo — BSac) 
= PIS? EI an rte G tenn, (2) 
=2Veo , ; ; ‘ (3) 
VOL. XXIX. PART II. VUE 
