382 



HAMILTON'S CHARACTERISTIC FUNCTION 



Let the position of the first point be referred to rectangular axes, the 

 origin of which is at P, and the axis of z, drawn in the direction PO. The 

 axes of x, and y, may be turned at pleasure round that of z, into the position 

 most suitable for our calculations. 



Let the position of the second point be referred to Q as origin, to QR 

 as axis of z,, and to axes of ic, and y, the position of which is of course 

 independent of that chosen for x lt y,. 



Let the ray from the first point & fa, y lt z,) to the second point R fa, y,, z,) 

 pass through P f (f l , 17,, 0), and <?(&, 17,, 0). 



We have then V ffK = V CfP .+ Vp, y + V yK 



Here 



and 

 Also 



(2). 

 (3), 

 .(4). 



Lif 



^ 



(5). 



+ terms involving higher powers and products of ,, 77,, ,, 1 



Writing, for the sake of brevity, single symbols for the differential coefficients, 

 we obtain for the value of V ffg up to the terms of the second degree inclusive 





+ 517,77, 



This is the value of V ffP + V ry + V yKt supposing the course of the ray to be 

 broken at P and <7 in an arbitrary manner. 



