292 Mr. L. Fletcher on the Dilatation of Crystals 
Neglecting squares of small quantities, 
tan POL'-tan POL = (8-S 1 ) tan POL ; 
or, to the same approximation, 
POI/-POL = (8-Sy) sin POL cos POL. 
Since 
POL' - POL = POP - 1/ OL, 
we may write the relation thus, 
P'OP-L / OL=(S-S 1 )sinPOLcosPOL . . . (A) 
From the above formula (Prop. VI.), 
P'OP = & sin Pa sin Pa= - [cos ua— cos(Pa + Pa)]. 
By Proposition XII. the motion of the crystal-line OL is 
the same as that of the plane-normal OL, for OL is a line of 
greatest or least expansion. 
Hence 
L'OL = £sin La sin La = - [cos aa— cos (La + La)], 
and 
P'OP-L'OL = | [cos (La + La) -cos (Pa + Pa)] 
. . Pa — La + Pa — La . Pa + La + Pa + La 
= k sin „ sin 
"Z "Z 
= k sin PL sin (PL + La + La). 
But, by Prop. XL (d), 
sin (La + La) =+1. 
■H.OI1CO 
FOP-L'OL=+£sinPLcosPL. . . . (B) 
Equating the two values given by (A) and (B), we find k 
is equal to +(S — 8i), and is independent of the particular 
pair of isotropic poles a, a. employed in the formula 6 = 
k sin Pa sin Pa. 
The positive or negative sign must be taken according as 
sin (La + La) is equal to +1 or —1, OL being that axis for 
which the coefficient of expansion is 8. 
Prop. XIV. To find the coefficient of expansion A of a line 
OP inclined to a direction of greatest or least expansion at an 
angle 6 (fig. 5). 
Let S, h x be the coefficients of expansion of the lines OL, OM, 
and let it be required to find the coefficient of expansion of 
OP where POL = 0. 
If P'N', N'O be the rectangular coordinates of P', the second 
