contained in the Third Book of the Mécanique Céleste. 35 
this purpose we must recollect that the expression we are con- 
sidering is a double fluent depending on two variable quan- 
tities. Let the variable quantities be functions of and ¢; then 
the element of the surface of the sphere will be dy sin dg, 
and the expression may be thus written, 
i ; 3 
Se +2a <<) dy sin fa’ (y'— y) d¢. 
Now, the integral {(y'— y) d¢ being taken between the limits 
¢=0 and $ = 27, it will be a function of cos /; and it is 
sufficient for the demonstration that this function be divisibie 
by the evanescent divisor. By this procedure the utmost ex- 
tent possible is given to the theorem; and after all, it will be 
found that we have obtained nothing but what readily follows 
from usual rules of analysis. 
1 
1 1 
= 2 
p= s=—+5- Co + <=. C°?) + &e. 
Let @ or — be expanded into a series, viz, 
the symbols C\), C\*), &c. being functions of cos : then 
by substituting this series in the function on the right-hand 
side of the formula (4), that function will become 
=: {= /y di sinflda! += .3 (Cy di sin de! + &e. (5) 
and by making a =r, we shall obtain 
y= Z x } fyas sin fda! +3f Cy'di sin fda’ +&e.t (6) 
Now the series (5) converges when a is less than 7; and 
therefore, even when it goes on ad infinitum, it may represent 
a finite quantity to any degree of approximation. But when 
a =7, the principle of convergency disappears, and no exact 
notion can be formed of the value of any finite number of the 
terms. It cannot therefore be said, with any precision of ideas, 
that such a series, consisting of an infinite number of terms 
without convergency, will represent any finite quantity. The 
mind cannot take in the whole series; it must be content with 
a definite portion of it; and no portion can be considered as 
equal to the quantity from which the whole is derived. It is 
only when the series breaks off, and consists of an assignable 
number of terms, that it can be said to represent a given quan- 
tity in the extreme case when a =7. Now this happens only 
when 7/ belongs to a certain class of functions ; namely, when 
it is a finite and rational function of three rectangular co- 
ordinates, which likewise comprehends every case in which the 
formula (4) is strictly demonstrated. For all such functions 
the equation (6) is exact, the two sides being identical, and 
differing from one another in nothing, except in the arrange- 
E2 ment 
