ELEMENTARY PRINCIPLES OF QUATERNIONS. 183 
We have to equate ¢, to ¢, and ¢, to ¢,, and then find the conditions of 
equality of the members of the equations. 
We do not change the results of the deductions if we equate simul- 
taneousl 
Y $,— $, tO 6; — Fy 
$, + G, to $, + G,. 
The first equation so formed will be (1.)— 
(1.) Tp Ta sin po [Up Uc — Uc Up] 
= (jk — ky) (bz — cy) + (ki — th) (ce — az) + (Y — Jr) (ay — ba). 
For the second equation let us put 
Up Uc + Uc Up=2F 
UE ip I kt He Oe a Pe = Dt”, 
This will give to it the following form (II.), after suppression of the 
factor 2— 
LS SN 
(I1.) Tp Ta[(Up)? cos px + F sin pa] 
= (Vax + poy +:F’cz) 
+ (bz + cy) f+ (cw + az) £" + (ay + bx) f”. 
Let us examine equation (I.) 
The coefficients (bz — cy), (cx — az), (ay — bx) are proportional to the 
cosines of the angles which the direction of Ur, defined in the preceding para- 
graph, forms with the three axes, z, y, z Namely, a perpendicular Ur to the 
plane of the directions Up, Us, drawn through a common origin, would be 
determined by the equations— 
ON “N INS 
a@ cos ar + b cos yr + € COS 27 = O 
ZN ZN N 
“cosar+ycosy7 + zcosz7=0. 
These give to the above coefficients the values comprised in 

bz —c¢ w—az ay —.bx , 
eas = ~ =TpTa sin po. 
A TX 
COS ar COS YT COS 27 
The sign of the last number has been taken so as to give the system of 
values (as a particular case) following— 
: 5 “~N 
p=mda,o=jy, cosz7r=+1. 
The condition (I.) may therefore be transformed into 
. “NW ° ° ee ee 
Up Uo — Uc Up = (jk — fy) cos ar + (ki — th) cos Yr + (4% —jt) cos ur 
VOL, XXVII. PART II. 3B 
