60 Professor Hamitton’s Third Supplement 
and are deduced by differentiation from the analogous equations of the first order 
} Slate oV oz ove 8V & 
Past f 
OS ACS oye 3 23 Cs + be wy" @ 
And the eleven equations thus deduced, when combined with the ten given by (W‘), 
and with the seven into which (7) resolves itself, suffice, in general, to determine 
the twenty-eight coefficients of V7, of the second order. 
Changes produced by Transformation of Co-ordinates. Nearly all the foregoing 
Results may be extended to Oblique Co-ordinates. The Fundamental Formula 
may be presented so as to extend even to Polar or any other marks of position ; and 
new Auxiliary Functions may then be found, analogous to, and including, the 
Functions W, T': together with New and General Differential and Integral 
Equations for Curved and Polygon Rays, Ordinary or Extraordinary. 
13. In all the foregoing investigations, it has been supposed that the final and 
initial co-ordinates, x, y, 2, wv’, y', 2, were referred to one common set of rectangular 
axes. But since it may be often convenient to change the mode of marking the final 
and initial positions, let us now express the old rectangular co-ordinates as linear 
functions of new and more general co-ordinates w,, ¥, 2,5 and w/, y/, 2’, which may 
or may not be rectangular, and may or may not be referred to one common set of 
final or initial axes. For this purpose we may employ the following formule, 
= Z, ar Ux, v, ais vy, Y, a Uz, Z, > ) 
Y=Yo + Ye, U,+Yy, Y, +Yz, 2,5 
o + Se, x, + Zy Y +22, 2,3 
(A*) 
aaa +0'e @) + ay Y +g! 23 
y =y,+y'z! z, +Y'yY, +¥Ye) B53 
hee) £5 ee wae , GR 
2=Z,+ 22) UL tZy Y, +2, z3 J 
in which each of the eighteen coefficients of the form 2, is the cosine of the angle 
between the directions of the two corresponding semiaxes, so that these coefficients are 
connected by the six following relations, on account of the rectangularity of the old 
co-ordinates, 
2 ° t 
Kx, + Yx, “=p Sx, 2— 1 5 x,” +Y a? + Z of hl 5 
2 
Gy, + Uy +4 2H13 ys tye? +2y2=1;3 (B*) 
3 2 2 
Tepe +Yz," +Zz = 1 5 ' x! +y'2/ ar Ze! elle 
