on Change of Temperature. 423 
mined by the substitution of the values of r, not in a single 
pair but in each of the three pairs of equations, the mean 
result being taken as the true value. 
For brevity, the angle (ir— PX) is denoted by PX, and 
thus —cos PX by cos PX. 
(a) To find U", the axis corresponding to the root + 3658 - 2. 
Substituting for r the value + 3658*2, and solving for 
x : y : z or cos L //7 X : cos X/"Y : cos U"Z, we get from the 
respective pairs of equations (1, 2), (2, 3), (3, 1): — 
(1,2) x:y:z 
(2, 3) x : y : z 
(3, 1) x :y :z 
whence 
logcosL"'X. 
(1.2) 9-9822662-1, 
(2.3) 9-9822692-7, 
(3,1) 9-9822665-0, 
: 1859510-1 
: 984541-9 
: 1527380-4 
-206053-4 : 501777-22, 
-109098-6 : 265644-3, 
-169354-64 : 412106-8 ; 
log cos L'" Y. 
9-0268474-4, 
9-0268542-4, 
9-0271163-7, 
logcosL'"Z. 
9-4133786-1, 
9-4133355-4, 
9-4133290-3: 
whence, taking the mean, we have from the equations (1, 2, 3) :- 
log cos L'"X, 9-9822673-3; log cos L'"Y, 9-0269393-5; 
log cos L'"Z, 9-4133477-3. 
(IS) To find M'", the axis corresponding to the root +31. 
Proceeding as before, we find 
(1.2) x:y 
(2.3) x:y 
(3,1) x:y 
whence _ 
logcosM'"X. 
(1.2) 8-9003438-5 
(2.3) 8-9005463-7 
(3,1) 8-9004600-8 
And for mean, 
(1,2,3) 8-9004501-0 
-88193 : 
-11381 : 
-112114 : 1104518 : 869134; 
869134 : 683819, 
112114 : 88193, 
log cos M'"Y. 
9-8939964-5 
9-8940258-5 
9-8939729-7 
logcosM'"Z. 
9-7898509-1 
9-7898000-7 
9-7898869-0 
9-8939984-2 9-7898459-6; 
(c) To find W", the axis corresponding to the root — 465-2. 
Proceeding as before, we find 
(1,2) x:y:z 
(2, 3) x : y : z 
(3,1) x:y:z 
204251-6 
73751-12 
336396-4 
465827-60 
168198-20 
767247-64 
-565689-92, 
-204251-60, 
-931655-20, 
