no Mr. Ivory on the expansion 
which equation is true when we make r — a. This analysis 
is equivalent to the demonstration of M. Poisson. Whatever 
may be thought of the reasoning, it cannot be denied that, in 
both processes, y' is in fact treated as a constant quantity. 
The equation is true of each individual molecule taken sepa- 
rately, and merely because its thickness has some determinate 
value. Such a demonstration cannot therefore be employed 
to prove that the thickness of a series of molecules covering 
the surface of a sphere, or a part of that surface, follows a 
certain law of variation, or comes under a particular deve- 
lopement. 
But a legitimate process of reasoning requires that, in the 
formula (A), while a represents any determinate quantity 
less than r, y' be considered as a function of the variable 
quantities sin 0', cos 0', sin q >' , cos <?>' ; and likewise that the 
integration be extended to the whole surface of the sphere, 
or to that part of it covered with the related molecules ; after 
which the true value of the formula will be obtained by 
making a = r. The whole system of molecules being com- 
prehended in the result, we may thence deduce, by a reverse 
process, the law according to which their thickness must vary, 
in order to produce that result. Now the integration here 
spoken of, cannot be executed, unless in the case when y' is 
explicitly a function of three rectangular co-ordinates. It is 
therefore only in this case that the differential equation can be 
considered as rigorously proved ; and it is remarkable, that, 
when we seek from that equation the developement of y 1 , it 
always comes out in a function of three rectangular co- 
ordinates. 
Whenjy' is not explicitly a function of three rectangular 
