799] 
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799. 
ON CURVILINEAR COORDINATES. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xix. (1883), 
pp. 1—22.] 
The present memoir is based upon Mr Warren’s “ Exercises in Curvilinear and 
Normal Coordinates,” Camb. Phil. Trans, t. xil. (1877), pp. 455—502, and has for 
a principal object the establishment of the six differential equations of the second 
order corresponding to his six equations for normal coordinates: but the notation is 
different; the results are more general, inasmuch as I use throughout general curvilinear 
coordinates instead of his normal coordinates; and as regards my six equations for 
general curvilinear coordinates, the terms containing differential coefficients of the first 
order are presented under a different form. 
If the position of a point in space is determined by the rectangular coordinates 
x, y, z; then p, q, r being each of them a given function of x, y, z, we have con 
versely x, y, z, each of them a given function of p, q, r, which are thus in effect 
coordinates serving to determine the position of the point, and are called curvilinear 
coordinates. 
But it is not in the first instance necessary to regard x, y, z as rectangular 
coordinates, or even as Cartesian coordinates at all; we are simply concerned with the 
two sets of variables x, y, z and p, q, r, each variable of the one set being a given 
function of the variables of the other set; and, in particular, the x, y, z are regarded 
as being each of them a given function of the p, q, r. 
Except as regards the symbols £, y, £ presently mentioned, the suffixes 1, 2, 3 refer 
to the variables p, q, r respectively, and are used to denote differentiations in regard 
to these variables, viz. 
/y> np rp rp rp rp 
^lj ^2, ^11 y tAj J2y *^22y ••• 
C. XIL 
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