Sir Witx1AM Rowan Hamitton’s Researches respecting Quaternions. 289 
“In my former letter I gave a theorem equivalent to that which I have here given as 
the theorem of the spherical triangle, answering, in fact, very nearly to the polar triangle 
conjugate therewith, but, as I think, much less geometrically simple, because the three 
corners had no obvious geometrical meanings, whereas now the corners R, R’, R” mark 
the directions of the factors and product respectively. In the new triangle, if we let fall a 
perpendicular from the extremity r” of that radius of the sphere which coincides in direc- 
tion with 7”, on the are Rr’, which represents the inclination of the factors to each other, 
and call the foot of this perpendicular r/’, we shall have 
“pt Par Bates erat 5 
TT COSA R is i eee SINR Bs (30) 
“ 
also the spherical coordinates of r,” will be ¢,”, ¥/"; and @,”, b,”, in (27), will be the 
spherical coordinates of a point x,” which will be one pole of the are rr’, and will be dis- 
tinguished from the other pole by the rule of rotation already assigned ; it might, perhaps, 
be called the positive pole of rr’, though it ought then to be considered as the negative 
pole of r’r. 
“* We saw that 7,” was in the plane of r and 7”, and this is now constructed by r,” being 
on the great circle RR’. 
«There seem to be some advantages in considering the quaternion 
win,” + jy," + he," (31) 
as the reduced product of the two factors already often mentioned in this letter; it 
is the part of their complete product (4) which is independent of their order ; and its 
radius 7”, is, as we have seen, the projection of the radius r” of the complete product on 
the plane of the two factors 77’. We now see that 
tan 9 sinrr/’= tan 0 sin?’ = tan 7/7’; (32) 
the radius 7/’ of the reduced product divides the angle between the radii r, 7’, of the 
factors, into parts, of which the sines are inversely as the tangents of the amplitudes, 0, 6’. 
Indeed this radius, 7/’, is the statical resultant, or algebraical sum, of two lines which 
coincide in direction with 7 and 7’ respectively, if w’ and w be positive, but have their 
lengths equal to the products w’r and wr’, or yp’ sin 8 cos 6 and jy’ sin &’ cos 8, or ww’ tan 6 
and ww’ tan §’; as appears (among other ways) from the equations (7). For the same 
reason, or by a combination of the equations (7), (16), (27), we have 
r/?u-p'— = cos 6 sin 8” + cos 9” sin 6? +2 sin 6 cos 8 sin 6’ cos @ cos rr’ ; (33) 
and because, by (21), 
cos 6” = cos 9 cos §’—sin @ sin @ cos 77’, (34) 
we arrive at the following pretty simple expression for the radius of the reduced product, 
r/'= pp’ ¥ (cos + cos 0”—2 cos 6 cos & cos 0”). (35) 
2Q2 
