346 Mr. L. Fletcher on the Dilatation of Crystals 
(4) Let P, E be the expansions of the unit lines in the 
directions OX, OC, and let E be the angle of rotation of the 
crystal-line OX for the given change of temperature; then 
E = g-1. 1 
For the point X we have, by writing /» = OX, and z=0 
in equations (II.), 
_, OX' . A'sin/3' , 
p = ox- 1= A^r 1; 
and, neglecting small quantities of the second order, 
_ XX' _ A'cos/3' _ C' cos/3 
OX ~ A sin (5 C sin 0" 
(5) In exactly the same way in which the relations of the 
rectangular coordinates £, £ of any point at the second tempe- 
rature to those of the same point at the first were found in 
terms of the alterations of the crystal-lines OA, 00, and of the 
angle AOC, we can find these coordinates in terms of the altera- 
tions of the crystal-lines OX, 00, and of the angle XOC; in 
fact we need only to write in equations (II.) ^-=- for - — 
XOC for AOC, and X'OC for A'OC. The relations may be 
determined also by substituting in equations (II.) the values 
of the coefficients given by equations (III.). 
We thus get, neglecting squares of small quantities, 
£=(1 + P)*, 
r=(i+R)*+Ea: 
.}•••• (iv.) 
or, reversing, 
— l 1 -™ I . . . . (V) 
(6) We can now find a formula for the determination of the 
lines of greatest and least expansion in either of the following 
ways : — 
First Method. — Let n/r, -^ be the inclinations of the same 
crystal-line OP, OP' to the line OX at the two temperatures ; 
then, from equations (IV.), 
tan^ = | = (1 | 1 R + ) ^ =:(l + R-P)tan^ + E, 
whence tan\/r' — tani/r = (R — P) tan yJr + E; 
and, still neglecting squares of small quantities, 
yfr'— f = (R-P) sin-^cos ^ + E cos 2 ^. 
