of Edinburgh, Session 1874-75. 
591 
any vector may be expressed in terms of any three other conter- 
minal and not complanar vectors, we may write 
<r — xa' + y/3' + zy' . 
Operate now by S.a. Then noticing that 
Sa /3' = 0 Say r = 0 
we have 
Sa <r~ = ccSaa' , 
cos AQ = x cos AA' , 
i.e., 
_ cos E 
sin p 1 * 
Similarly 
_ cos E _ cos E 
^ sin p 2 * ^ sin p i 
Hence 
er 
= cos E ( E— + X- + 
V sm p x Sin p 2 
Or since 
Y /3y = a' sin a , &c., 
we may write 
J-) 
mi pj 
cos E 
sin a sin 
Pi 
(V/3y + Yya + Y a/3) 
Or we might proceed thns- 
Since 
QA = QB = QC , 
Sera = S<r/3 = Scry, 
Str (a - (3) = 0 , S cr(j3 - y) = 0 , 
<r is J_ r plane of chordal A . 
cr = zY Q3y + ya + a/3) . 
Operate by Sa. Then 
Sacr = zSaY/3y = 2 sin aSaa , 
cos E 
sin a sin p l ’ 
therefore 
therefore 
therefore 
Hence 
therefore 
(5.) 
(5.) 
z = 
