104 Mr. Ivory on the expansion 
of three rectangular co-ordinates, and when it must be made 
to assume the form of such a function by a transformation. 
When y is an explicit function of three rectangular co-or- 
dinates, these things are true : 
ist. The developement contains no quantities except what 
are found in y. The two expressions are entirely equivalent, 
being in reality the same quantities differently arranged. 
2dly. Wheny is a finite expression, the developement will 
consist of a finite number of terms ; and wheny is a converg- 
ing series of an infinite number of terms, the developement 
will be a like converging series. For, in the case of a con- 
verging series, we may approach to the value of y as near as 
we please, by taking in a determinate number of the terms ; 
and the developement of this portion of the series will like- 
wise consist of a finite number of terms. 
On the other hand, the properties that demand attention 
are very different, when we suppose that y is not explicitly a 
function of three rectangular co-ordinates. 
ist. The developement will always contain an infinite num- 
ber of terms. 
adly. In most instances; and, more particularly, in the 
very general example that has been considered above ; the 
terms of the developement will involve an infinite number of 
quantities which do not appear in the original function, and 
which are introduced merely in order to give the develope- 
ment its peculiar form. 
The original function and the whole infinite series of which 
the developement consists, may be represented by this equa- 
tion, viz. 
y =y -J- M — M : 
