108 Mr. Ivory on the expansion 
to a particular law, or being expressed by some function of the 
arcs that determine the position of a molecule : the differen- 
tial equation will be true of all these molecules. But it will 
be equally true of them, if the thickness be entirely arbitrary, 
and subject to no law of variation. It is true, in reality, of 
each molecule taken in an insulated manner, and because its 
thickness is some determinate quantity. How then can such 
an equation be a fit means of proving that the thickness 
varies in one way rather than another, or that it comes under 
a particular developement ? 
To consider this matter more particularly, lety =/(0, <p) 
denote, as before, the thickness of the molecule ; and put, 
/=/(«', ¥). 
y = cos 9 cos 0'-|- sin 9 sin 9' cos ( <p — <p * ) , 
^ = V y — 2 ra . y + a*, 
a being less than r: then tt denoting the semi-circumference 
to the radius i, the differential equation at the surface is what 
the following formula becomes in the particular case ofr=<z, 
viz. 
<*•7! 
y = ii + 2a ~drh' s inffdfdifi 
or, by substituting the value of 
y 4 jy (r*— a*) ?/ sin 6' dV d<p' . 
( r* — 2 ray + a 1 ) 2 
the integral being taken between the limits 0'=o, <p / = o and 
G^tt, <p'=27r, and making a=r, after the integration. 
Now, when r—a, r* — a 2 =o; and in this case all the ele- 
ments of the integral are evanescent, unless in the particular 
circumstances when the denominator is evanescent, or when 
