Measurements of Wave-Lengths. 557 
kept up to the distance-pieces, led to a calculation on Hertz's 
theory o£ the relation between the change of interval and 
the pressure. If two spheres of radii r x and r 2 and of 
material for which the elastic constants in Lame's notation 
are X 1? /a 1? \ 2 , /j, 2 , are pressed together with a force P, the 
relation between P and the distance (a) through which the 
centres approach one another, as the result of the deforma- 
tion in the neighbourhood of the contact, is 
P = " 
where 
4 / r 1 r 2 \» a * 
3Tr\n+7 2 ) e 1 +e 2 ' 
e _ \! + 2/x 1 e= \ 2 + 2 f j J2 
1 47r/Lt 1 (\ 1 + At 1 )' 2 ±Tr{JL 2 (\ 2 + fjL 2 )' 
In the case of materials which satisfy Poisson's condition, 
\ = /a, and we may take as sufficiently approximate 
3 3 
87r/x x ' " 87r/x 2 ' 
so that 
__ 32 / i\r 2 y- ^i^a«* 
In the application that we have to make, one of the spheres 
is of steel (invar) and of radius r, ='25 cm., while the other 
is of glass and of radius r 2 = co . Further, for the steel we 
may take /,6 1 = 8 , 2 x 10 u , and for glass ju 2 = 2'4 x 10 11 , and 
thus 
P = 3*30 xlO 11 .*!, 
« being in cm. and P in dynes. It will be convenient for 
our purpose to reckon a in wave-lengths (equal say to 
6 x 10 — 5 cm.) and P in kilograms, taking the dyne as equal 
to a milligram weight. On this understanding 
P = -15 at 
signifying that to cause an approach of one wave-length the 
force required is '15 kilogram. If P and a undergo small 
* See Love's Math. Theory of Elasticity, § 139. 
Phil. Mag. S. 6. Vol. 15. No. ^. April 1908. 2 P 
