431 
and then, © being an prbibiary function of (a y, 2), or of 
Geen wy. 
in) , 
ae 
then (a, ...), (A, ...) are given functions of the differential 
: dx du : : 
coefficients is CF6a5 coo OE ae &e., that is of (2, y, 2), or, what is 
Cy. (BYaanana mee. wy 
the same thing, of (u, v, w), such that 
eee hy gare On tues fel tem se Gh, Eb 
=be—f? : ca—g’ : ab—h® : gh—af : hf—bg : fg—ch, 
and 
Amc rab h sis f oa Breese ee ce 
—BO-F?: CA-@: AB-H’ : GH-AF: HF-BG@ : FG—CH, 
and the theory of curvilinear co-ordinates is in fact a theory of 
the mutual relations of these coefficients (a, ... ) and (A, ... ). 
In Lamé’s system of curvilinear co-ordinates where the 
surfaces u=0, v=0, w=0 are orthotomic, f= ¢=h=0, and 
therefore also F= G@=H=O0: and the remaining coefficients 
correspond to Lamé’s h, h,, h,, H, H,, H,; viz. we have 
Lee dap Pa ae = 
gi Boge eye 
and Lamé gives six differential equations of the second order 
satisfied by h, h,, h,, or 4, H,, H,, considered as functions of the 
variables which correspond to (u, v, w). fd 
In the author’s system of normal co-ordinates, u,v, w denote 
the normal distances of the point (a, y, z) from three given 
surfaces w= 0, v=0, w=0 respectively: and the coefficients 
are then such that d= B= C=1. He obtains on this assump- 
