ON THE ALGEBRAIC COUPLE. . 185 
which has not failed to attract the attention of eminent mathematicians ; and 
- [trust I may be permitted to avail myself of this Meeting of the British Asso- 
ciation in Dublin, as a convenient opportunity for the publication of the 
views which I had in contemplation on the occasion of my former paper. 
These views having suggested themselves in the course of a brief investiga- 
tion relating to the interpretation of the Algebraic Couple, I propose to in- 
troduce this subject also, in the hope that it may prove interesting to those 
who have given their attention to the various systems of Multiple Algebra 
which have been from time to time propounded. 
On the Geometrical Interpretation of the Algebraic Couple. 
The object of this section is to apply and interpret the Algebraic Couple 
to and by means of the geometry of angular magnitude and position. 
_ The couple in its ordinary form, #+y / —1, is the argument of the arbi- 
trary function, f(at+y “—1), which represents a value of w in the partial 
differential equation 
Tu, du 
dai" dy 
_ If we take the corresponding differential equation of three variables, 
Gu Gu, tu_ 
dete dies aa 
and effect its integration, not generally, but under the restrictive condition 
#+y?+2°=r (a constant), 
we obtain : 
u=o(tan- Y+tan-? ——) ; 
xv r 
_ whiéh may be regarded as an integration of the equation upon the surface of 
_ asphere whose radius is 7. If 7/ be the longitude of a point computed from 
any origin, and ) its distance in latitude from the equator, the integral 
assumes the form 
vA - => i+ thd ) 
4 . of =o cos A)’ 
5 or . 
_., u=o(UF V —1 log tans), 
where yp is the co-latitude. The argument of this function is now in the 
form of an ordinary algebraic couple, the constituents of which are an- 
gular magnitudes; and my object will be to show that the couple in this 
_ form is an adequate symbolical representation of position on a sphere, or of 
_ angular position in space, in the same manner as the ordinary couple ade- 
quately represents position on a plane. 
It will be convenient for the sake of comparison to consider the algebraic 
couple, when geometrically interpreted, rather as an operation, than as a 
"quantity or result. Let us regard «+y%/—I1 not merely as denoting the 
position of a point (x, y), but implying also the process of arriving at such a 
point from the origin by progressing along an unvarying course, viz. that 
course which is constantly inclined to the unit-line at an angle whose tan- 
